AP CALCULUS AB

Chapter 5:The Definite Integral

Section 5.1:Estimating with Finite Sums

What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output

… and whyLearning about estimating with finite sums

sets the foundation for understanding integral calculus.

Section 5.1 – Estimating with Finite Sums Distance Traveled at a Constant Velocity:

A train moves along a track at a steady rate of 75 mph from 2 pm to 5 pm. What is the total distance traveled by the train?

2 5

75mph

t

v(t)

TDT = Area under line = 3(75) = 225 miles

Section 5.1 – Estimating with Finite Sums Distance Traveled at Non-Constant Velocity:

75

2 5 8t

v(t)

Total Distance Traveled = Area of geometric figure = (1/2)h(b1+b2) = (1/2)75(3+8) = 412.5 miles

Example Finding Distance Traveled when Velocity Varies

2A particle starts at 0 and moves along the -axis with velocity ( )

for time 0. Where is the particle at 3?

x x v t t

t t

Graph and partition the time interval into subintervals of length . If you use

1/ 4, you will have 12 subintervals. The area of each rectangle approximates

the distance traveled over the subint

v t

t

erval. Adding all of the areas (distances)

gives an approximation to the total area under the curve (total distance traveled)

from 0 to 3.t t

Example Finding Distance Traveled when Velocity Varies

2

Continuing in this manner, derive the area 1/ 4 for each subinterval and

add them:

1 9 25 49 81 121 169 225 289 361 441 529 2300

256 256 256 256 256 256 256 256 256 256 256 256 2568.98

im

Example Estimating Area Under the Graph of a Nonnegative Function

2Estimate the area under the graph of ( ) sin from 0 to 3.f x x x x x

Applying LRAM on a graphing calculator using 1000 subintervals, we find the left endpoint approximate area of 5.77476.

Section 5.1 – Estimating with Finite Sums Rectangular Approximation Method

15

5 secLower Sum = Area ofinscribed = s(n)

Upper Sum = Areaof circumscribed= S(n)

Midpoint Sum

n

ii xxfA

1

sigma = sum y-value at xi

width of region

nSns region of Area

LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3

Section 5.1 – Estimating with Finite Sums

Rectangular Approximation Method (RAM) (from Finney book)

1 2 3

y=x2

LRAM = Left-hand Rectangular Approximation Method

= sum of (height)(width) of each rectangleheight is measured on left side of

each rectangle

875.6

2

1

2

5

2

12

2

1

2

3

2

11

2

1

2

1

2

10

22

22

22

LRAM

Section 5.1 – Estimating with Finite Sums Rectangular Approximation Method (cont.)

y=x2

RRAM = Right-hand RectangularApproximation Method

= sum of (height)(width) of eachrectangle

height is measured on right sideof rectangle

375.11

2

13

2

1

2

5

2

12

2

1

2

3

2

11

2

1

2

1 22

22

22

RRAM

1 2 3

Section 5.1 – Estimating with Finite Sums Rectangular Approximation Method (cont.)

y=x2

1 2 3

MRAM = Midpoint RectangularApproximation Method

= sum of areas of each rectangleheight is determined by the heightat the midpoint of each horizontal region

9375.8

2

1

4

11

2

1

4

9

2

1

4

7

2

1

4

5

2

1

4

3

2

1

4

1222222

MRAM

Section 5.1 – Estimating with Finite Sums Estimating the Volume of a Sphere

The volume of a sphere can be estimated by a similar method using the sum of the volume of a finite number of circular cylinders.

definite_integrals.pdf (Slides 64, 65)

Section 5.1 – Estimating with Finite Sums Cardiac Output problems involve the

injection of dye into a vein, and monitoring the concentration of dye over time to measure a patient’s “cardiac output,” the number of liters of blood the heart pumps over a period of time.

Section 5.1 – Estimating with Finite Sums See the graph below. Because the

function is not known, this is an application of finite sums. When the function is known, we have a more accurate method for determining the area under the curve, or volume of a symmetric solid.

Section 5.1 – Estimating with Finite Sums Sigma Notation (from Larson book)

The sum of n terms is written as

is the index of summationis the ith term of the sum

and the upper and lower bounds of summation are n and 1 respectively.

naaaa ,...,,, 321

n

ini aaaaa

1321 ...

iai

Section 5.1 – Estimating with Finite Sums Examples:

1...1312111

54321

2222

1

2

5

1

ni

i

n

i

i

Section 5.1 – Estimating with Finite Sums Properties of Summation

1.

2.

n

i

n

iii akka

1 1

n

i

n

i

n

iiiii baba

1 1 1

Section 5.1 – Estimating with Finite Sums Summation Formulas:

1.

2.

3.

4.

n

i

cnc1

n

i

nni

1 2

1

n

i

nnni

1

2

6

121

n

i

nni

1

223

4

1

Section 5.1 – Estimating with Finite Sums Example:

3080

115121252

1110

4

1211002

11010

4

11010

1

22

10

1

10

1

3

10

1

10

1

32

i i

i i

ii

iiii

Section 5.1 – Estimating with Finite Sums Limit of the Lower and Upper Sum

If f is continuous and non-negative on the interval [a, b], the limits as of both the lower and upper sums exist and are equal to each other

l.subinterva on the of valuesmaximum and

minimum theare and and where

limlimlimlim

th

1 1

if

Mfmfn

abx

nSxMfxmfns

ii

n

i

n

in

in

inn

n

Section 5.1 – Estimating with Finite Sums Definition of the Area of a Region in the Plane

Let f be continuous an non-negative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is

n

abx

xcxxcf iii

n

ii

n

and

,lim Area 11 (ci, f(ci))

xi-1 xi