1 − x^2 = 1 − x
x^2 − x = 0
x(x − 1) = 0
x = 0, 1The left-hand edge of the region is x = 0 and the right-hand edge is x = 1, so the limits of integration are
from 0 to 1.
Next, note that the top curve is always y = 1 − x^2 , and the bottom curve is always y = 1 − x. (If the region
has a place where the top and bottom curve switch, you need to make two integrals, one for each region.
Fortunately, that’s not the case here.) Thus, we need to evaluate the following:
Sometimes you’re given the endpoints of the region; sometimes you have to find them on your own.
Example 2: Find the area of the region between the curve y = sin x and the curve y = cos x from 0 to .
First, sketch the region.