MAT 1235 Calculus II 4.1, 4.2 Part I The Definite Integral

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Presentation transcript:

MAT 1235 Calculus II 4.1, 4.2 Part I The Definite Integral

Homework WebAssign HW 4.2 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

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

Preview

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 i th subinterval sample point

In General

i th subinterval sample point

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

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

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

In General Sum of the area of the rectangles is

Definition

upper limit lower limit integrand

Definition 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

Example 2

Example 3 Compute by interpreting it in terms of area

Example 4 Compute

Properties The follow properties are labeled according to the textbook.

Property (a)

Example 5

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

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 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!