lesson 23: the definite integral (slides)
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Section 5.2The Definite Integral
V63.0121.006/016, Calculus I
New York University
April 15, 2010
Announcements
I April 16: Quiz 4 on §§4.1–4.4I April 29: Movie Day!!I April 30: Quiz 5 on §§5.1–5.4I Monday, May 10, 12:00noon (not 10:00am as previouslyannounced) Final Exam
. . . . . .
. . . . . .
Announcements
I April 16: Quiz 4 on§§4.1–4.4
I April 29: Movie Day!!I April 30: Quiz 5 on§§5.1–5.4
I Monday, May 10,12:00noon (not 10:00amas previously announced)Final Exam
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 2 / 28
. . . . . .
Objectives
I Compute the definiteintegral using a limit ofRiemann sums
I Estimate the definiteintegral using a Riemannsum (e.g., Midpoint Rule)
I Reason with the definiteintegral using itselementary properties.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 3 / 28
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 4 / 28
. . . . . .
Cavalieri's method in general
Let f be a positive function defined on the interval [a,b]. We want tofind the area between x = a, x = b, y = 0, and y = f(x).For each positive integer n, divide up the interval into n pieces. Then
∆x =b− an
. For each i between 1 and n, let xi be the ith step betweena and b. So
. .x..x0
..x1
..xi
..xn−1
..xn.. . . .. . .
x0 = a
x1 = x0 +∆x = a+b− an
x2 = x1 +∆x = a+ 2 · b− an
. . .
xi = a+ i · b− an
. . .
xn = a+ n · b− an
= b
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 5 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
left endpoints…
Ln =n∑
i=1
f(xi−1)∆x
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
right endpoints…
Rn =n∑
i=1
f(xi)∆x
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
midpoints…
Mn =n∑
i=1
f(xi−1 + xi
2
)∆x
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
the minimum value on theinterval…
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
the maximum value on theinterval…
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
…even random points!
. .x. . . . . . .
In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Forming Riemann sums
We have many choices of representative points to approximate thearea in each subinterval.
…even random points!
. .x. . . . . . .In general, choose ci to be a point in the ith interval [xi−1, xi]. Form theRiemann sum
Sn = f(c1)∆x+ f(c2)∆x+ · · ·+ f(cn)∆x =n∑
i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L1 = 3.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L2 = 5.25
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L3 = 6.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L4 = 6.375
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L5 = 6.59988
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L6 = 6.75
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L7 = 6.85692
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L8 = 6.9375
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L9 = 6.99985
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L10 = 7.04958
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L11 = 7.09064
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L12 = 7.125
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L13 = 7.15332
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L14 = 7.17819
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L15 = 7.19977
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L16 = 7.21875
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L17 = 7.23508
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L18 = 7.24927
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L19 = 7.26228
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L20 = 7.27443
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L21 = 7.28532
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L22 = 7.29448
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L23 = 7.30406
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L24 = 7.3125
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L25 = 7.31944
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L26 = 7.32559
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L27 = 7.33199
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L28 = 7.33798
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L29 = 7.34372
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.left endpoints
.L30 = 7.34882
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R1 = 12.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R2 = 9.75
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R3 = 9.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R4 = 8.625
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R5 = 8.39969
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R6 = 8.25
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R7 = 8.14236
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R8 = 8.0625
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R9 = 7.99974
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R10 = 7.94933
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R11 = 7.90868
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R12 = 7.875
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R13 = 7.84541
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R14 = 7.8209
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R15 = 7.7997
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R16 = 7.78125
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R17 = 7.76443
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R18 = 7.74907
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R19 = 7.73572
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R20 = 7.7243
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R21 = 7.7138
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R22 = 7.70335
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R23 = 7.69531
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R24 = 7.6875
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R25 = 7.67934
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R26 = 7.6715
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R27 = 7.66508
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R28 = 7.6592
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R29 = 7.65388
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.right endpoints
.R30 = 7.64864
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M1 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M2 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M3 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M4 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M5 = 7.4998
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M6 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M7 = 7.4996
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M8 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M9 = 7.49977
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M10 = 7.49947
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M11 = 7.49966
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M12 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M13 = 7.49937
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M14 = 7.49954
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M15 = 7.49968
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M16 = 7.49988
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M17 = 7.49974
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M18 = 7.49916
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M19 = 7.49898
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M20 = 7.4994
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M21 = 7.49951
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M22 = 7.49889
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M23 = 7.49962
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M24 = 7.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M25 = 7.49939
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M26 = 7.49847
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M27 = 7.4985
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M28 = 7.4986
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M29 = 7.49878
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M30 = 7.49872
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U1 = 12.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U2 = 10.55685
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U3 = 10.0379
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U4 = 9.41515
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U5 = 8.96004
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U6 = 8.76895
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U7 = 8.6033
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U8 = 8.45757
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U9 = 8.34564
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U10 = 8.27084
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U11 = 8.20132
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U12 = 8.13838
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U13 = 8.0916
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U14 = 8.05139
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U15 = 8.01364
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U16 = 7.98056
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U17 = 7.9539
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U18 = 7.92815
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U19 = 7.90414
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U20 = 7.88504
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U21 = 7.86737
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U22 = 7.84958
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U23 = 7.83463
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U24 = 7.82187
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U25 = 7.80824
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U26 = 7.79504
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U27 = 7.78429
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U28 = 7.77443
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U29 = 7.76495
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.maximum points
.U30 = 7.7558
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L1 = 3.0
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L2 = 4.44312
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L3 = 4.96208
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L4 = 5.58484
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L5 = 6.0395
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L6 = 6.23103
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L7 = 6.39577
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L8 = 6.54242
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L9 = 6.65381
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L10 = 6.72797
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L11 = 6.7979
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L12 = 6.8616
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L13 = 6.90704
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L14 = 6.94762
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L15 = 6.98575
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L16 = 7.01942
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L17 = 7.04536
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L18 = 7.07005
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L19 = 7.09364
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L20 = 7.1136
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L21 = 7.13155
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L22 = 7.14804
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L23 = 7.16441
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L24 = 7.17812
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L25 = 7.19025
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L26 = 7.2019
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L27 = 7.21265
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L28 = 7.22269
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L29 = 7.23251
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.minimum points
.L30 = 7.24162
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 8 / 28
. . . . . .
The definite integral as a limit
DefinitionIf f is a function defined on [a,b], the definite integral of f from a to bis the number ∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 9 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrandI a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrandI a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrand
I a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrandI a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrandI a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
Notation/Terminology
∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci)∆x
I∫
— integral sign (swoopy S)
I f(x) — integrandI a and b — limits of integration (a is the lower limit and b theupper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)I The process of computing an integral is called integration orquadrature
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 10 / 28
. . . . . .
The limit can be simplified
TheoremIf f is continuous on [a,b] or if f has only finitely many jumpdiscontinuities, then f is integrable on [a,b]; that is, the definite integral∫ b
af(x) dx exists.
TheoremIf f is integrable on [a,b] then∫ b
af(x)dx = lim
n→∞
n∑i=1
f(xi)∆x,
where∆x =
b− an
and xi = a+ i∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 11 / 28
. . . . . .
The limit can be simplified
TheoremIf f is continuous on [a,b] or if f has only finitely many jumpdiscontinuities, then f is integrable on [a,b]; that is, the definite integral∫ b
af(x) dx exists.
TheoremIf f is integrable on [a,b] then∫ b
af(x)dx = lim
n→∞
n∑i=1
f(xi)∆x,
where∆x =
b− an
and xi = a+ i∆x
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 11 / 28
. . . . . .
Example: Integral of x
Example
Find∫ 3
0x dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)(3n
)=
9n2
n∑i=1
i
=9n2
· n(n+ 1)2
−→ 92
So∫ 3
0x dx =
92= 4.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 12 / 28
. . . . . .
Example: Integral of x
Example
Find∫ 3
0x dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)(3n
)=
9n2
n∑i=1
i
=9n2
· n(n+ 1)2
−→ 92
So∫ 3
0x dx =
92= 4.5
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 12 / 28
. . . . . .
Example: Integral of x2
Example
Find∫ 3
0x2 dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)2(3n
)=
27n3
n∑i=1
i2
=27n3
· n(n+ 1)(2n+ 1)6
−→ 273
= 9
So∫ 3
0x2 dx = 9
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 13 / 28
. . . . . .
Example: Integral of x2
Example
Find∫ 3
0x2 dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)2(3n
)=
27n3
n∑i=1
i2
=27n3
· n(n+ 1)(2n+ 1)6
−→ 273
= 9
So∫ 3
0x2 dx = 9
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 13 / 28
. . . . . .
Example: Integral of x3
Example
Find∫ 3
0x3 dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)3(3n
)=
81n4
n∑i=1
i3
=81n4
· n2(n+ 1)2
4−→ 81
4
So∫ 3
0x3 dx =
814
= 20.25
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 14 / 28
. . . . . .
Example: Integral of x3
Example
Find∫ 3
0x3 dx
Solution
For any n we have ∆x =3nand xi =
3in. So
Rn =n∑
i=1
f(xi)∆x =n∑
i=1
(3in
)3(3n
)=
81n4
n∑i=1
i3
=81n4
· n2(n+ 1)2
4−→ 81
4
So∫ 3
0x3 dx =
814
= 20.25
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 14 / 28
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 15 / 28
. . . . . .
Estimating the Definite Integral
Example
Estimate∫ 1
0
41+ x2
dx using M4.
Solution
We have x0 = 0, x1 =14, x2 =
12, x3 =
34, x4 = 1.
So c1 =18, c2 =
38, c3 =
58, c4 =
78.
M4 =14
(4
1+ (1/8)2+
41+ (3/8)2
+4
1+ (5/8)2+
41+ (7/8)2
)
=14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
6465
+6473
+6489
+64113
≈ 3.1468
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 16 / 28
. . . . . .
Estimating the Definite Integral
Example
Estimate∫ 1
0
41+ x2
dx using M4.
Solution
We have x0 = 0, x1 =14, x2 =
12, x3 =
34, x4 = 1.
So c1 =18, c2 =
38, c3 =
58, c4 =
78.
M4 =14
(4
1+ (1/8)2+
41+ (3/8)2
+4
1+ (5/8)2+
41+ (7/8)2
)
=14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
6465
+6473
+6489
+64113
≈ 3.1468
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 16 / 28
. . . . . .
Estimating the Definite Integral
Example
Estimate∫ 1
0
41+ x2
dx using M4.
Solution
We have x0 = 0, x1 =14, x2 =
12, x3 =
34, x4 = 1.
So c1 =18, c2 =
38, c3 =
58, c4 =
78.
M4 =14
(4
1+ (1/8)2+
41+ (3/8)2
+4
1+ (5/8)2+
41+ (7/8)2
)=
14
(4
65/64+
473/64
+4
89/64+
4113/64
)
=6465
+6473
+6489
+64113
≈ 3.1468
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 16 / 28
. . . . . .
Estimating the Definite Integral
Example
Estimate∫ 1
0
41+ x2
dx using M4.
Solution
We have x0 = 0, x1 =14, x2 =
12, x3 =
34, x4 = 1.
So c1 =18, c2 =
38, c3 =
58, c4 =
78.
M4 =14
(4
1+ (1/8)2+
41+ (3/8)2
+4
1+ (5/8)2+
41+ (7/8)2
)=
14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
6465
+6473
+6489
+64113
≈ 3.1468
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 16 / 28
. . . . . .
Estimating the Definite Integral (Continued)
Example
Estimate∫ 1
0
41+ x2
dx using L4 and R4
Answer
L4 =14
(4
1+ (0)2+
41+ (1/4)2
+4
1+ (1/2)2+
41+ (3/4)2
)= 1+
1617
+45+
1625
≈ 3.38118
R4 =14
(4
1+ (1/4)2+
41+ (1/2)2
+4
1+ (3/4)2+
41+ (1)2
)=
1617
+45+
1625
+12≈ 2.88118
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 17 / 28
. . . . . .
Estimating the Definite Integral (Continued)
Example
Estimate∫ 1
0
41+ x2
dx using L4 and R4
Answer
L4 =14
(4
1+ (0)2+
41+ (1/4)2
+4
1+ (1/2)2+
41+ (3/4)2
)= 1+
1617
+45+
1625
≈ 3.38118
R4 =14
(4
1+ (1/4)2+
41+ (1/2)2
+4
1+ (3/4)2+
41+ (1)2
)=
1617
+45+
1625
+12≈ 2.88118
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 17 / 28
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 18 / 28
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a,b] and c a constant. Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx+
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x)− g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 19 / 28
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a,b] and c a constant. Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx+
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x)− g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 19 / 28
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a,b] and c a constant. Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx+
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x)− g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 19 / 28
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a,b] and c a constant. Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx+
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x)− g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 19 / 28
. . . . . .
Proofs
Proofs.
I When integrating a constant function c, each Riemann sumequals c(b− a).
I A Riemann sum for f+ g equals a Riemann sum for f plus aRiemann sum for g. Using the sum rule for limits, the integral of asum is the sum of the integrals.
I Ditto for constant multiplesI Ditto for differences
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 20 / 28
. . . . . .
Example
Find∫ 3
0
(x3 − 4.5x2 + 5.5x+ 1
)dx
Solution
∫ 3
0(x3−4.5x2 + 5.5x+ 1)dx
=
∫ 3
0x3 dx− 4.5
∫ 3
0x2 dx+ 5.5
∫ 3
0x dx+
∫ 3
01 dx
= 20.25− 4.5 · 9+ 5.5 · 4.5+ 3 · 1 = 7.5
(This is the function we were estimating the integral of before)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 21 / 28
. . . . . .
Example
Find∫ 3
0
(x3 − 4.5x2 + 5.5x+ 1
)dx
Solution
∫ 3
0(x3−4.5x2 + 5.5x+ 1)dx
=
∫ 3
0x3 dx− 4.5
∫ 3
0x2 dx+ 5.5
∫ 3
0x dx+
∫ 3
01 dx
= 20.25− 4.5 · 9+ 5.5 · 4.5+ 3 · 1 = 7.5
(This is the function we were estimating the integral of before)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 21 / 28
. . . . . .
Example
Find∫ 3
0
(x3 − 4.5x2 + 5.5x+ 1
)dx
Solution
∫ 3
0(x3−4.5x2 + 5.5x+ 1)dx
=
∫ 3
0x3 dx− 4.5
∫ 3
0x2 dx+ 5.5
∫ 3
0x dx+
∫ 3
01 dx
= 20.25− 4.5 · 9+ 5.5 · 4.5+ 3 · 1 = 7.5
(This is the function we were estimating the integral of before)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 21 / 28
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a,b]or has finitely many jumpdiscontinuities, then
limn→∞
Sn = limn→∞
{ n∑i=1
f(ci)∆x
}
exists and is the same value nomatter what choice of ci we make. .
.
.
.x.midpoints
.M15 = 7.49968
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 22 / 28
. . . . . .
More Properties of the Integral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
This allows us to have
5.∫ c
af(x)dx =
∫ b
af(x)dx+
∫ c
bf(x)dx for all a, b, and c.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 23 / 28
. . . . . .
More Properties of the Integral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
This allows us to have
5.∫ c
af(x)dx =
∫ b
af(x)dx+
∫ c
bf(x)dx for all a, b, and c.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 23 / 28
. . . . . .
More Properties of the Integral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
This allows us to have
5.∫ c
af(x)dx =
∫ b
af(x)dx+
∫ c
bf(x)dx for all a, b, and c.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 23 / 28
. . . . . .
Example
Suppose f and g are functions with
I∫ 4
0f(x)dx = 4
I∫ 5
0f(x)dx = 7
I∫ 5
0g(x)dx = 3.
Find
(a)∫ 5
0[2f(x)− g(x)] dx
(b)∫ 5
4f(x)dx.
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 24 / 28
. . . . . .
SolutionWe have(a) ∫ 5
0[2f(x)− g(x)] dx = 2
∫ 5
0f(x)dx−
∫ 5
0g(x)dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f(x)dx =
∫ 5
0f(x)dx−
∫ 4
0f(x)dx
= 7− 4 = 3
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 25 / 28
. . . . . .
SolutionWe have(a) ∫ 5
0[2f(x)− g(x)] dx = 2
∫ 5
0f(x)dx−
∫ 5
0g(x)dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f(x)dx =
∫ 5
0f(x)dx−
∫ 4
0f(x)dx
= 7− 4 = 3
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 25 / 28
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 26 / 28
. . . . . .
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a,b].
6. If f(x) ≥ 0 for all x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) for all x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M for all x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 27 / 28
. . . . . .
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a,b].6. If f(x) ≥ 0 for all x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) for all x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M for all x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 27 / 28
. . . . . .
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a,b].6. If f(x) ≥ 0 for all x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) for all x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M for all x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 27 / 28
. . . . . .
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a,b].6. If f(x) ≥ 0 for all x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) for all x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M for all x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 27 / 28
. . . . . .
Example
Estimate∫ 2
1
1xdx using the comparison properties.
SolutionSince
12≤ x ≤ 1
1for all x in [1,2], we have
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 28 / 28
. . . . . .
Example
Estimate∫ 2
1
1xdx using the comparison properties.
SolutionSince
12≤ x ≤ 1
1for all x in [1,2], we have
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1
V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 28 / 28
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