1. Definite Integrals
Day 5

2. So what have we learned?
We have learned that if you integrate a function, you find an expression to determine the area under that curve.
f(x)

3. What if we have two curves?
What would the area between these two curves be?
f(x)
g(x) A

4. How about these two curves?
From a to b, what is true about
Is the equation going to change?
f(x)
g(x) A

5. Lets look at some animations
This is a Riemann Sum Area between curves

6. Many problems here will ask you to find the area between two curves without defining a and b. This means you have to find a and b.
If both equations are y in terms of x, set them equal and solve for x. Then do a quick sketch. THEN and only then pull out your calculator to make sure you are correct, and continue.

7. Why can’t I always use my calculator?
Because on tests and on a good portion of the AP exam you won’t be able to use your calculator anyway. Tomorrow we’ll learn how to integrate on the calculator. BUT as this tool won’t be available to you on tests, you need to understand and be able to do this work. The calculator simply checks your work for you.

8. Examples
Find the area between the curves: