5.2 Part I The Definite Integral MAT 1235 Calculus II 5.2 Part I The Definite Integral http://myhome.spu.edu/lauw

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.)

Major Themes in Calculus I

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

Major Themes in Calculus II

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

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

Example 0

Example 0 Use left hand end points to get an estimation

Example 0 Use right hand end points to get an estimation

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

In General 𝑖th subinterval sample point

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

In General ith subinterval sample point

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

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

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

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

Definition The Definite Integral of 𝑓 from 𝑎 to 𝑏

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

Definition The Definite Integral of 𝑓 from 𝑎 to 𝑏 Integration : Process of computing integrals

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

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

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.

More Remarks If 𝑓(𝑥)≥0 on [𝑎,𝑏], then

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

More Remarks

Example 2

Example 3 Compute by interpreting it in terms of area

Example 4 Compute

Properties

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

Example 5

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

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

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

Example 7 If , what is ?

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

Property (c)

Example 8

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.