4.5 (part 2) Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002

Objectives Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function.

The technique is a little different for definite integrals. Example: The technique is a little different for definite integrals. new limit We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits.

Wrong! The limits don’t match! Example: Using the original limits: Leave the limits out until you substitute back. Wrong! The limits don’t match! This is usually more work than finding new limits

Example: new limit new limit Find new limits.

Example: Don’t forget to use the new limits.

Example:

Integration of Even and Odd Functions Even if f(-x)=f(x) Symmetric with respect to y-axis Odd if f(-x)=-f(x) Symmetric with respect to origin

If f is an even function then If f is an odd function then

Odd

In another generation or so, we might be able to use the calculator to find all integrals. Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators. You must practice finding integrals by hand until you are good at it! p

Homework 4.5 (page 297) #41-93 odd