limits to y-limits. The two curves intersect at y = , so our limits of integration are from y = 0 to y =
. The new integral is
You get the same answer no matter which way you integrate (as long as you do it right!). The challenge of
area problems is determining which way to integrate and then converting the equation to different terms.
Unfortunately, there’s no simple rule for how to do this. You have to look at the region and figure out its
endpoints, as well as where the curves are with respect to each other.
Once you can do that, then the actual set-up of the integral(s) isn’t that hard. Sometimes, evaluating the
integrals isn’t easy; however, if the integral of an AP question is difficult to evaluate, you’ll be required
only to set it up, not to evaluate it.
Here are some sample problems. On each, decide the best way to set up the integrals, and then evaluate
them. Then check your answer.
PROBLEM 1. Find the area of the region between the curve y = 3 − x^2 and the line y = 1 − x from x = 0 to x
= 2.
Answer: First, make a sketch.