1. 5.2 Part I The Definite Integral
MAT 1235 Calculus II
5.2 Part I
The Definite Integral

2. Homework WebAssign HW 5.2 I
Review the Closed Interval Method for part II (Section 4.2)
(“I do not remember”, or “I have never learned it” are not options.)

3. Major Themes in Calculus I

4. Major Themes in Calculus I
We do not like to use the definition
Develop techniques to deal with different functions

5. Major Themes in Calculus II

6. Major Themes in Calculus II
We do not like to use the definition
Develop techniques to deal with different functions

7. Preview
Look at the definition of the definite integral 𝑦=𝑓(𝑥) on [𝑎,𝑏]
Look at its relationship with the area between the graph 𝑦=𝑓(𝑥) and the 𝑥-axis on [𝑎,𝑏]
Properties of Definite Integrals

8. Example 0

9. Example 0
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
𝑖th subinterval
sample point

13. In General
Suppose 𝑓 is a continuous function defined on [𝑎,𝑏], we divide the interval [𝑎,𝑏] into 𝑛 subintervals of equal width
The area of the 𝑖𝑡ℎ rectangle is

14. In General
ith subinterval
sample point

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

16. Sigma Notation for summation
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
Final value (upper limit)
Initial value (lower limit)
Index

18. In General Sum of the area of the rectangles is
As we increase 𝑛, we get better and better estimations.

19. Definition
The Definite Integral of 𝑓 from 𝑎 to 𝑏

20. Definition The Definite Integral of 𝑓 from 𝑎 to 𝑏 upper limit
lower limit
integrand

21. Definition The Definite Integral of 𝑓 from 𝑎 to 𝑏
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
If 𝑓(𝑥)≥0 on [𝑎,𝑏], then

26. More Remarks
If 𝑓(𝑥)≥0 on [𝑎,𝑏], then
If 𝑓(𝑥)≤0 on [𝑎,𝑏], then

27. More Remarks

28. Example 2

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

30. Example 4
Compute

31. Properties

32. Property (a)
𝑥, 𝑡 are called the dummy variables

33. Example 5

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

35. upper limit < lower limit
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 (do problem #2)
The one sit with you in the same table IS your partner! Work with your partner and your partner ONLY.
Persons who give away their answers will be penalized. Keep your voice down.
Once you get checked, you can go.
Classwork must be finished within the class time to count.