These next two units discuss some of the most difficult topics you’ll encounter in AP Calculus. For some
reason, students have terrible trouble setting up these problems. Fortunately, the AP Exam asks only
relatively simple versions of these problems on the exam.
Unfortunately, this unit and the next are always on the AP Exam. We’ll try to make them as simple as
possible. You’ve already learned that if you want to find the area under a curve, you can integrate the
function of the curve by using the endpoints as limits. So far, though, we’ve talked only about the area
between a curve and the x-axis. What if you have to find the area between two curves?
VERTICAL SLICES
Suppose you wanted to find the area between the curve y = x and the curve y = x^2 from x = 2 to x = 4.
First, sketch the curves.
You can find the area by slicing up the region vertically, into a bunch of infinitely thin strips, and adding
up the areas of all the strips. The height of each strip is x^2 − x, and the width of each strip is dx. Add up
all the strips by using the integral.
(x^2 − x) dxThen, evaluate it.