1. MAT 1235 Calculus II 4.1, 4.2 Part I The Definite Integral http://myhome.spu.edu/lauw

2. Homework WebAssign HW 4.2 I

3. Major Themes in Calculus I

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

5. Major Themes in Calculus II

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

7. Preview

8. Example 0

9. Use left hand end points to get an estimation

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

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

12. In General i th subinterval sample point

13. In General

14. i th subinterval sample point

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

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

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

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

19. Definition

20. upper limit lower limit integrand

21. Definition Integration : Process of computing integrals

22. Example 1 Express the limit as a definite integral on the given interval.

23. Example 1 Express the limit as a definite integral on the given interval.

24. Remarks We are not going to use this limit definition to compute definite integrals. In section 4.3, we are going to use antiderivative (indefinite integral) to compute definite integrals. We will use this limit definition to derive important properties for definite integrals.

25. More Remarks

28. Example 2

29. Example 3 Compute by interpreting it in terms of area

30. Example 4 Compute

31. Properties The follow properties are labeled according to the textbook.

32. Property (a)

33. Example 5

34. Property (b) The definition of definite integral is well- defined even if upper limit < lower limit And

35. Property (b) The definition of definite integral is well- defined even if upper limit < lower limit And

36. Example 6 Note: If lower limit > upper limit, the integral has no obvious geometric meaning

37. Example 7 If, what is ?

38. Example 7 If, what is ? Q1: What is the answer? Q2: How many steps are needed to clearly demonstrate the solutions?

39. Property (c)

40. Example 8

41. Classwork 2 persons per group. Work with your partner and your partner ONLY. Once you get checked, you can go. Please take a cookie on your way out!