4.3 riemann sums and definite integrals. the definite integral in the section 4.2, the definition of...
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4.3 Riemann Sums and Definite Integrals
The Definite Integral
In the Section 4.2, the definition of area is defined as
The Definite Integral
The following example shows that it is not necessary to have subintervals of equal widthExample 1 Find the area bounded by the graph of
and x-axis over the interval [0, 1].
SolutionLet ( i = 1, 2, …, n) be the endpoint of the subinteravls. Then the width of the i th subinterval is
2
2
2
2 )1(
n
i
n
ixi
22
22 12)1(
n
i
n
ii
The width of all subintervals varies.Let ( i = 1, 2, …, n) be the point in the i th subinteravls, then
ii xc
2
2
n
ixi
xxf )(
So, the limit of sum is
Continued…
Example 1 Find the area bounded by the graph of
and x-axis over the interval [0, 1].
Solution
i
n
ii
nxcf
1
)(lim
2)( xxf
Let and ( i = 1, 2, …, n) be the endpoint of the subinteravls and the point in the i th subinterval.
2
2
n
ixi ii xc
2
12
2 12lim
n
i
n
in
in
n
in
iin 1
23
21
lim
2
)1(
6
)12)(1(2
1lim
3
nnnnn
nn
3
21lim
12
11
3
1lim
2
n
n
nn nn
Definition of a Riemann Sum
= partition of [a, b]
1 iii xxx
= length of the i th subintervalNorm of |||| = length of the longest subinterval
b
adxxf )(
“definite integral of f from a to b ”
definition
0|||| 1
)( lim
n
iii xcf
Riemann Sum - approximates the definite integral
area, f(x) > 0 on [a, b]
net area, otherwise
The Definite Integral
a
b
Upper Limit
Lower Limit
b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrandvariable of integration
(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
Definition of a Definite Integral
Theorem 4.4 Continuity Implies Integrability
Questions
(1)Is the converse of Theorem 4.4 true? Why? (2)If change the condition of Theorem 4.4 “f
is continuous” to “f is differentiable”, is the Theorem 4.4 true?
(3)Of the conditions “continuity”, “differentiability” and “integrability”, which one is the strongest?
Answers
(1)False. Counterexample is
(2)Yes. Because “f is differentiable” implies “f is continuous”
(3)The order from strongest to weakest is “integrability”, “continuity”, and “differentiability”.
About Theorem 4.4 Continuity Implies Integrability
)(xf1, when x ≠ 1 on [0, 5]0, otherwise
f
a b
A
Adxxfb
a )(
a b
fA1
A2
A3
231)( AAAdxxfb
a
= area above – area below
The Definite Integral
n
n
i n
abi
n
abaf
1 lim
If using subintervals of equal length, (regular partition), with ci chosen as the right endpoint of the i th subinterval, then
b
adxxf )(
Regular Right-Endpoint Formula (RR-EF)
Special Cases
n
n
i n
abi
n
abaf
1)1( lim
If using subintervals of equal length, (regular partition), with ci chosen as the left endpoint of the i th subinterval, then
b
adxxf )(
Regular Left-Endpoint Formula (RL-EF)
Special Cases
f
a
adxxf )( 0 by definition
a b
a
bdxxf )(
b
adxxf )( by definition
b
adxxf )(
c
adxxf )(
b
cdxxf )(
c
Theorem 4.6 Properties of the Definite Integral
Theorem 4.6 Properties of the Definite Integral
2. 0a
af x dx If the upper and lower limits are equal,
then the integral is zero.
1. b a
a bf x dx f x dx Reversing the limits
changes the sign.
b b
a ak f x dx k f x dx 3. Constant multiples can be
moved outside.
b b b
a a af x g x dx f x dx g x dx 4.
Integrals can be added and subtracted.
b
adxxkf )(
b
adxxfk )(
b
adxxgxf )()(
b
a
b
adxxgdxxf )()(
Theorem 4.7 Properties of the Definite Integral
Example 2 If
3)(5
2 dxxf and ,10)(
9
2 dxxf
then find
.)(45
9 dxxf
Examples
5
9)(4 dxxf
Solution
9
5)(4 dxxf
9
2
5
2)()(4 dxxfdxxf
3104 28
HomeworkPg. 278 9, 13-19 odd, 25-31 odd, 33-41 odd, 45-49, 55