1. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 5 Review of Basic Integration Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

2. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 5.2.1 (p. 322) Antiderivatives

3. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Table 5.2.1 (p. 324)

4. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.2.3 (p. 325)

5. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Equations (4) – (7) (p. 326)

6. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.2.1 (p. 327) Antiderivatives = Family of Functions

7. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 1. Evaluate a) b) c) d) Question 1

8. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 5.1.1 (p. 317) The Area Problem The Area Problem

9. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.1.4 (p. 318)

10. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.4.4 (p. 344) Rectangular Approximation for Area = This is called a “Riemann Sum” for the area.

11. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 5.4.3 (p. 344) Area Under a Curve Definition of Area for a continuous function

12. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.1.4 (p. 318)

13. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.4.10 (p. 349) Riemann sums yield “signed” area

14. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.4.7 (p. 347) Three Common Approximations

15. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.5.1 (p. 353) Approximations with variable width

16. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 5.5.1 (p. 354) Definite Integral

17. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.5.2 (p. 355) Continuous Functions are Integrable

18. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 5.5.3 (p. 356) Definite Integral Rules

19. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.5.4 (p. 357) Definite Integral Rules

20. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.5.5 (p. 358) Definite Integral Rules

21. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral vs Antiderivatives We have seen two basic ideas so far: Antiderivative: Computes a family of functions Definite Integral: Computes a number = area Is there a connection between these two?

22. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.6.1 (p. 363) The Fundamental Theorem of Calculus, Part 1 Fundamental Theorem of Calculus - I

23. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Fundamental Theorem of Calculus - I Practice

24. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 5.6.69 (p. 375)

25. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.6.3 (p. 370) The Fundamental Theorem of Calculus, Part 2 Fundamental Theorem of Calculus - II

26. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Guidelines for u-Substitution (p. 334) U-Substitution

27. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 2. Evaluate a) 89.5 b) 97.5 c) 96.5 d) -89.5 Question 2

28. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 3. Find the area under the curve over the interval [9, 16]. a) 91 b) c) 37 d) Question 3

29. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 4. Use part 2 of the Fundamental Theorem of Calculus to find: a) b) c) d) Question 4

30. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 6. Evaluate a) b) c) d) Question 6

31. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.c 2.a 3.b 4.d 5. 6.a Answers