Objectives for Section 13.5 Fundamental Theorem of Calculus

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 Barnett/Ziegler/Byleen Business Calculus 11e

3. Evaluating Definite Integrals By the fundamental theorem we can evaluate easily and exactly. We simply calculate Barnett/Ziegler/Byleen Business Calculus 11e

4. Definite Integral Properties Barnett/Ziegler/Byleen Business Calculus 11e

5. Example 1 Make a drawing to confirm your answer. 0  x  4 - 1  y  6 Barnett/Ziegler/Byleen Business Calculus 11e

6. Example 2 Make a drawing to confirm your answer. 0 x 4 - 1 y 4 Barnett/Ziegler/Byleen Business Calculus 11e

7. Example 3 0 x 4 - 2 y 10 Barnett/Ziegler/Byleen Business Calculus 11e

8. Example 4 Let u = 2x, du = 2 dx Barnett/Ziegler/Byleen Business Calculus 11e

9. Example 5 Barnett/Ziegler/Byleen Business Calculus 11e

10. Example 6 This is a combination of the previous three problems Barnett/Ziegler/Byleen Business Calculus 11e

11. Example 7 Let u = x3 + 4, du = 3x2dx Barnett/Ziegler/Byleen Business Calculus 11e

12. 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: Barnett/Ziegler/Byleen Business Calculus 11e

13. 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 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. 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 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. Example • Section 6.5 #70. 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). Barnett/Ziegler/Byleen Business Calculus 11e

18. Example(continued) a) Find the average cost per unit if 1000 dictionaries are produced Solution: The average cost is Barnett/Ziegler/Byleen Business Calculus 11e

19. Example(continued) b) Find the average value of the cost function over the interval [0, 1000] Solution: 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 evaluate a definite integral by the fundamental theorem of calculus: We can find the average value of a function f by Barnett/Ziegler/Byleen Business Calculus 11e