Fundamental Theorem of Calculus: Evaluating Definite Integrals and Average Values

1. Chapter 6Integration 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. Evaluating Definite Integrals By the fundamental theorem we can evaluate easily and exactly. We simply calculate

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 = x3 + 4, du = 3x2dx

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: 0 ≤x≤ 3 –1 ≤y≤ 3 Example 7: –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. 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. Example • The total cost (in dollars) of printing x dictionaries is C(x) = 20,000 + 10x • Find the average cost per unit if 1000 dictionaries are produced. • Find the average value of the cost function over the interval [0, 1000]. • Write a description of the difference between part a) and part b).

19. Example(continued) a) Find the average cost per unit if 1000 dictionaries are produced Solution: The average cost is

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 evaluate a definite integral by the fundamental theorem of calculus: We can find the average value of a function f by