MAT 1235 Calculus II 4 2 Part I

MAT 1235 Calculus II 4. 2 Part I The Definite Integral http: //myhome. spu. edu/lauw

Homework l l l Web. Assign HW 5. 2 I Review the Closed Interval Method for part II (see MAT 1234 Section 3. 1) (“I do not remember”, or “I have never learned it” are not options. )

Major Themes in Calculus I

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

Preview l

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 sample point

In General l

In General 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 Final value (upper limit) Index Initial value (lower limit)

In General Sum of the area of the rectangles is

Definition l

Definition l upper limit lower limit integrand

Definition l 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 l l l 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

More Remarks l

More Remarks

Example 2

Example 3 Compute terms of area by interpreting it in

Example 4 Compute

Properties

Property (a) l

Example 5

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

Property (b) The definition of definite integral is welldefined 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 ? Q 1: What is the answer? Q 2: How many steps are needed to clearly demonstrate the solutions?

Property (c)

Example 8

Classwork (do problem #2) l l l 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.