1. MAT 1221 Survey of Calculus Section 6.4 Area and the Fundamental Theorem of Calculus http://myhome.spu.edu/lauw

2. Quiz 8 minutes

3. Major Themes in Calculus

4. We do not like to use the definition Develop techniques to deal with different functions

5. Major Themes in Calculus

6. We do not like to use the definition Develop techniques to deal with different functions

7. Preview

8. Key Pay attention to the overall ideas Pay less attention to the details – We are going to use a formula to compute the definite integrals, not limits.

9. Example 0

10. Use left hand end points to get an estimation

11. Example 0 Use right hand end points to get an estimation

12. Example 0 Observation: What happen to the estimation if we increase the number of subintervals?

13. In General i th subinterval sample point

14. In General

15. sample point

16. In General Sum of the area of the rectangles is Riemann Sum

17. In General Sum of the area of the rectangles is Sigma Notation for summation

18. In General Sum of the area of the rectangles is Index Initial value (lower limit) Final value (upper limit)

19. In General Sum of the area of the rectangles is

20. Definition

21. upper limit lower limit integrand

22. Definition Integration : Process of computing integrals

23. Remarks We are not going to use this limit definition to compute definite integrals. We are going to use antiderivative (indefinite integral) to compute definite integrals.

24. Area and Indefinite Integrals

25. Otherwise, the definite integral may not have obvious geometric meaning.

26. Example 1 Compute by interpreting it in terms of area.

27. Example 1 We are going to use this example to verify our next formula.

28. Fundamental Theorem of Calculus

29. Remarks To simplify the computations, we always use the antiderivative with C=0.

30. Remarks To simplify the computations, we always use the antiderivative with C=0. We will use the following notation to stand for F(b)-F(a):

31. FTC

32. Example 2

33. Example 3

34. Example 4

35. The Substitution Rule for Definite Integrals For complicated integrands, we use a version of the substitution rule.

36. The Substitution Rule for Definite Integrals The procedures for indefinite and definite integrals are similar but different. We need to change the upper and lower limits when using a substitution. Do not change back to the original variable.

37. The Substitution Rule for Definite Integrals

39. Example 5

40. Example 6

41. Physical Meanings of Definite Integrals We will not have time to discuss the exact physical meanings. Basic Idea: The definite integral of rate of change is the net change.

42. Example 7 (HW 18)