Objectives for Section 13.5 Fundamental Theorem of Calculus

Barnett/Ziegler/Byleen Business Calculus 11e 1

Objectives for Section 13.5 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.


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


By the fundamental theorem we can evaluate

easily and exactly. We simply calculate

Evaluating Definite Integrals

b

adxxf )(

)()( aFbF


Definite Integral Properties

a

adxxf 0)(

a

b

b

adxxfdxxf )()(

b

a

b

adxxfkdxxfk )()(

b

a

b

a

b

adxxgdxxfdxxgxf )()(])()([

b

c

c

a

b

adxxfdxxfdxxf )()()(


Example 1

Make a drawing to confirm your answer.

105151535553

1

3

1

xxdx

0 x 4

- 1 y 6


Example 2

421

29

2

3

1

23

1

x

xdxx

Make a drawing to confirm your answer.

0 x 4

- 1 y 4


Example 3

9093

3

0

33

0

2 x

xdxx

0 x 4- 2 y 10


Example 4

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

1

2 dxe x

3.6268604222

221

221

1

2

1

1

1

1

eee

edue

x

x

x

u

x

u


Example 5

0.69314718 2ln 1ln – 2ln

ln1 2

1

2

1

x

xdxx


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


Example 7

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

5

0 3

2

dx

xx

1.1578393 4)/3 (ln– 129)/3(ln

3)4(ln

3ln1

31

43

31

5

0

3

5

0

5

0

5

0 3

2

x

xx

x

uduu

dxx

x


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

3ln

xx

xu


Numerical Integration on a Graphing Calculator

Use some of the examples from previous slides:

2

1

1 dxx

5

0 3

2

4dx

xx

Example 5:

Example 7:

0 x 3- 1 y 3

-1 x 6

- 0.2 y 0.5


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.


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

x

xxdxx

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:


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.


Section 6.5 #70. 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


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

Solution: The average cost is

Example(continued)

1020000)()( xx

xCxC

3010100020000)1000( C


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


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.


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

Documents

Objectives for Section 13.5 Fundamental Theorem of Calculus