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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.
Barnett/Ziegler/Byleen Business Calculus 11e 2
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
Barnett/Ziegler/Byleen Business Calculus 11e 3
By the fundamental theorem we can evaluate
easily and exactly. We simply calculate
Evaluating Definite Integrals
b
adxxf )(
)()( aFbF
Barnett/Ziegler/Byleen Business Calculus 11e 4
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 )()()(
Barnett/Ziegler/Byleen Business Calculus 11e 5
Example 1
Make a drawing to confirm your answer.
105151535553
1
3
1
x
xdx
0 x 4
- 1 y 6
Barnett/Ziegler/Byleen Business Calculus 11e 6
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
Barnett/Ziegler/Byleen Business Calculus 11e 7
Example 3
9093
3
0
33
0
2
x
xdxx
0 x 4
- 2 y 10
Barnett/Ziegler/Byleen Business Calculus 11e 8
Example 4
Let u = 2x, du = 2 dx?1
1
2 dxe x
3.6268604222
22
1
221
1
2
1
1
1
1
eee
edue
x
x
x
u
x
u
Barnett/Ziegler/Byleen Business Calculus 11e 9
Example 5
0.69314718 2ln 1ln – 2ln
ln1 2
1
2
1
xxdx
x
Barnett/Ziegler/Byleen Business Calculus 11e 10
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
Barnett/Ziegler/Byleen Business Calculus 11e 11
Example 7
Let u = x3 + 4, du = 3x2 dx?4
5
0 3
2
dx
x
x
1.1578393
4)/3 (ln– 129)/3(ln 3
)4(ln
3
ln1
3
1
4
3
3
1
5
0
3
5
0
5
0
5
0 3
2
x
xx
x
udu
udx
x
x
Barnett/Ziegler/Byleen Business Calculus 11e 12
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
Barnett/Ziegler/Byleen Business Calculus 11e 13
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
Barnett/Ziegler/Byleen Business Calculus 11e 14
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.
Barnett/Ziegler/Byleen Business Calculus 11e 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.
$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:
Barnett/Ziegler/Byleen Business Calculus 11e 16
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.
Barnett/Ziegler/Byleen Business Calculus 11e 17
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
Barnett/Ziegler/Byleen Business Calculus 11e 18
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
Barnett/Ziegler/Byleen Business Calculus 11e 19
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
Barnett/Ziegler/Byleen Business Calculus 11e 20
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.
Barnett/Ziegler/Byleen Business Calculus 11e 21
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
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