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

4 By the fundamental theorem we can evaluate easily and exactly. We simply calculate Evaluating Definite Integrals

5 Definite Integral Properties

6 Example 1 Make a drawing to confirm your answer. 0 ≤ x ≤ 4 –1 ≤ y ≤ 6

7 Example 2 Make a drawing to confirm your answer. 0 ≤ x ≤ 4 –1 ≤ y ≤ 4

8 Example 3 0 ≤ x ≤ 4 –2 ≤ y ≤ 10

9 Example 4 Let u = 2x, du = 2 dx

10 Example 5

11 Example 6 This is a combination of the previous three problems

12 Example 7 Let u = x 3 + 4, du = 3x 2 dx

13 Example 7 (revisited) 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:

14 Numerical Integration on a Graphing Calculator Use some of the examples from previous slides: 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) = 90x 2 + 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) = 90x 2 + 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. Solution:

17 Using Definite Integrals for Average Values The average value of a continuous function f over [a, b] is 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, x 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)

20 Example (continued) b) Find the average value of the cost function over the interval [0, 1000] Solution:

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: