1. 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

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

3. 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.

4. 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

5. Example:
new limit
new limit
Find new limits.

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

7. Example:

8. 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

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

10. Odd

11. 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

12. Homework
4.5 (page 297)
#41-93 odd