MAT 1235 Calculus II 4.2 Part II The Definite Integral

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Presentation transcript:

MAT 1235 Calculus II 4.2 Part II The Definite Integral

Office Hours MTRF3-4pm I have chocolate in my office!

Next WebAssign HW 4.2 II Quiz: 4.1, 4.2 You need to have a study group by Next Wednesday. Quiz question: List the names for your study group members

Review: Major Themes

Review: Motivation i th subinterval sample point

Review: Definite Integral

Review: Interpretations in terms of Areas

Review: Properties

Preview 8 Properties of Definite Integrals Two levels of understanding The properties make sense to you You know how to use the properties

Preview 8 Properties of Definite Integrals Two levels of understanding The properties make sense to you You know how to use the properties Most properties can be visualized using positive functions. They are true for all (continuous) functions even if the functions are not positive.

Preview We will use the Closed Interval Method (Calculus I)

Property #1

Property #2,3,4 (Linear Property)

Example 1 If, what is ? Q1: What is the answer? Q2: How many steps are needed to clearly demonstrate the solutions?

Example 1 If, what is ?

Property #5

Remarks The property is true even if upper limits < lower limits

Example 2 Simplify

Example 2 Simplify Q1: What is the answer? Q2: How many steps are needed to clearly demonstrate the solutions?

Example 2 Simplify

Property #6

Property #7 (Skip this if time...)

Property #7 Geometric Meaning

Example 3

It suffices to show that on,

Backward Engineering! It suffices to show that on,

Property #8 (Skip this if time...)

Property #8 Geometric Meaning

Example 4

Step 1: Use the Closed Interval Method to find M and m. Step 2: Use property 8 to find the estimations.

Recall The Close Interval Method

Idea

The Closed Interval Method

Step 1: Use the Closed Interval Method to find M and m

You are expected to give formal answers

Step 2: Use property 8 to find the estimations

Expectations Organize your solutions Clearly show how you find the critical number(s). (Make sure all critical numbers are within the interval) Clearly show the function values. Clearly state the abs max/min values. Range need to be specific at certain places. (Be sure to practice with your HW)