barrels Problem of the Day 200 100 6 12 18 24 hours The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day? A) 500 B) 600 C) 2,400 D) 3,000 E) 4,800

barrels Problem of the Day 200 100 6 12 18 24 hours The flow of oil, in barels per hour, through a pipeline on July 9 is given by the graph above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day? A) 500 B) 600 C) 2,400 D) 3,000 E) 4,800

Homework Questions?

4-5: Integration By Substitution Objectives: Learn and practice the substitution technique for integration Integrate even and odd functions ©2003 Roy L. Gover (www.mrgover.com)

Review Find the derivative:

Important Idea The role of substitution in integration is comparable to the chain rule in differentiation. Substitution, sometimes called u-substitution (u-sub), allows integration that otherwise could not be done.

Example Find the antiderivative: u Then: And: Let:

Procedure Let u=the inside function. Find du in terms of dx. Substitute and find the antiderivative in terms of u. Substitute the expression for x in place of u.

Example Find the antiderivative: u

Try This Find the antiderivative using a u-substitution then differentiate your result:

Warm-Up Solve the differential equation with the initial condition of y(0)=2:

Example Find the antiderivative

Important Idea The u-substitution must eliminate any product of variables.

Example Find the antiderivative:

Example Find the antiderivative: Ex 5, p295 This example requires a slightly different technique:

Try This Find the antiderivative: 301/46

Example Find the antiderivative: Ex6,p296

Try This Find the antiderivative:

Try This Find the antiderivative: Ex7a,p297

Example Find the antiderivative: Let u=? Ex 7b,p297

Example Find the antiderivative: Ex7c,p 297

Try This Find the antiderivative: Ex7e,p 297

Assignment 304/7-25 odd & 31-37odd(find the general solution), 43-49 odd Slides 1-21 2. 305/51-63 odd ,71,73 Slides 22-38

Warm-Up Find the antiderivative: Ex7d,p297

Important Idea Substitution methods are also used with definite integrals. There are two methods: Reverse the substitution and use original limits Change the variable and use new limits

Example Evaluate:

Try This 820

Example Evaluate:

Definition Odd functions are functions symmetric with the origin. They have exponents that are all odd.

Example Odd Function

Definition Even Functions are functions symmetric with the y axis. They have exponents that are all even. Constants are considered to have even exponents.

Example Even Function

Try This Even, odd or neither?

Important Idea If f is an even function:

Important Idea If f is an odd function:

Example Evaluate:

Example Evaluate:

Try This Evaluate:

Lesson Close Describe generally how you integrate a function by substitution.

Assignment 304/7-25 odd & 31-37odd(find the general solution), 43-49 odd Slides 1-21 2. 305/51-63 odd ,71,73 Slides 22-38