Chapter 6

Integration

Section 5

The Fundamental Theorem of Calculus

2

Objectives for Section 6.5 Fundamental Theorem of Calculus

■ The student will be able to evaluate definite integrals.

■ The student will be able to calculate the average value of a function using the definite integral.

3

Fundamental Theorem of Calculus

If f is a continuous function on the closed interval [a, b], and F is any antiderivative of f, then

)()()()( ba aFbFxFdxxf

b

a

4

By the fundamental theorem we can evaluate

easily and exactly. We simply calculate

Evaluating Definite Integrals

b

adxxf )(

)()( aFbF

5

Definite Integral Properties

[ f (x) g (x)] dx

a

b

f (x) dxa

b

g (x) dxa

b

f (x)dx

a

b

f (x)dxa

c

f (x)dxc

b

kf (x)dx

a

b

k f (x)dxa

b

f (x)dx a

b

f (x)dxb

a

f (x)dx 0a

a

6

Example 1

Make a drawing to confirm your answer.

105151535553

1

3

1

x

xdx

0 ≤ x ≤ 4

–1 ≤ y ≤ 6

7

Example 2

42

1

2

9

2

3

1

23

1

x

xdxx

Make a drawing to confirm your answer.

0 ≤ x ≤ 4

–1 ≤ y ≤ 4

8

Example 3

9093

3

0

33

0

2

x

xdxx

0 ≤ x ≤ 4

–2 ≤ y ≤ 10

9

Example 4

Let u = 2x, du = 2 dx?1

1

2 dxe x

1

2eu du

x 1

1

eu

2 x 1

1

e2x

2 x 1

1

e2

2

e 2

2

3.6268604

10

Example 5

0.69314718 2ln 1ln – 2ln

ln1 2

1

2

1

xxdx

x

11

Example 6

207.78515

1ln – )/2(e– 1/3– 3ln )/2(e 9

ln23

1

26

3

1

23

3

1

22

x

x

x

xex

dxx

ex

This is a combination of the previous three problems

12

Example 7

Let u = x3 + 4, du = 3x2 dx?4

5

0 3

2

dx

x

x

1

3

3x2

x3 4dx

0

5

1

3

1

udu

x0

5

ln u

3 x 0

5

ln(x3 4)

3 x 0

5

(ln 129)/3 Š (ln 4)/3

1.1578393

13

Example 7(revisited)

1.1578393 4)/3(ln – 129)/3(ln 3

ln 129

4

u

u

On the previous slide, we made the back substitution from u back to x. Instead, we could have just evaluated the definite integral in terms of u:

5

0

35

0 3

)4(ln

3

ln

xx

xu

14

Numerical Integration on a Graphing Calculator

Use some of the examples from previous slides:

2

1

1dx

x

5

0 3

2

4dx

x

x

Example 5:

Example 7:

0 ≤ x ≤ 3

–1 ≤ y ≤ 3

–1 ≤ x ≤ 6

–0.2 ≤ y ≤ 0.5

15

Example 8

From past records a management service determined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by M ´(x) = 90x2 + 5,000, where M(x) is the total accumulated cost of maintenance for x years.

Write a definite integral that will give the total maintenance cost from the end of the second year to the end of the seventh year. Evaluate the integral.

16

Example 8

From past records a management service determined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by M ´(x) = 90x2 + 5,000, where M(x) is the total accumulated cost of maintenance for x years.

$29,480 15,000– 810– 35,000 10,290

500030000,5907

3

37

3

2

xxxdxx

Write a definite integral that will give the total maintenance cost from the end of the second year to the end of the seventh year. Evaluate the integral.

Solution:

17

Using Definite Integrals for Average Values

The average value of a continuous function f over [a, b] is

dxxfab

b

a)(

1

Note this is the area under the curve divided by the width. Hence, the result is the average height or average value.

18

The total cost (in dollars) of printing x dictionaries is C(x) = 20,000 + 10x

a) Find the average cost per unit if 1000 dictionaries are produced.

b) Find the average value of the cost function over the interval [0, 1000].

c) Write a description of the difference between part a) and part b).

Example

19

a) Find the average cost per unit if 1000 dictionaries are produced

Solution: The average cost is

Example(continued)

1020000)(

)( xx

xCxC

30101000

20000)1000( C

20

Example(continued)

b) Find the average value of the cost function over the interval [0, 1000]

Solution:

25,000 5,000 20,000

)520000(1000

1

)1020000(1000

1)(

1

1000

0

2

1000

0

x

b

a

xx

dxxdxxfab

21

Example(continued)

c) Write a description of the difference between part a and part b

Solution: If you just do the set-up for printing, it costs $20,000. This is the cost for printing 0 dictionaries.

If you print 1,000 dictionaries, it costs $30,000. That is $30 per dictionary (part a).

If you print some random number of dictionaries (between 0 and 1000), on average it costs $25,000 (part b).

Those two numbers really have not much to do with one another.

22

Summary

We can find the average value of a function f by

We can evaluate a definite integral by the fundamental theorem of calculus:

)()()()( aFbFxFdxxf ba

b

a

dxxfab

b

a)(

1