Area between the curves?

Question

Find the area between the curves given by

x = -y^2 + 3y +4 and

x = y + 1 .

Answer 1

the way you find the area BETWEEN curves is to find the area UNDER one of them and subtract it from the area UNDER the other one. I.e., you integrate and subtract the integrals from each other, and take the absolute value of the difference. Don't be confused by the fact that x is in the usual role of y, and vice-versa. Both integrals are easy to calculate.

The next step is to figure out where the curves cross. That's when both RHSs equal each other, which boils down to y^2 - 2y- 3 = 0, which is a quadratic with roots 3 and -1. So those are your most important limits of integration.

There's one more step. Find the zeros of each of the two functions that occur between y = -1 and y = 3. Those are relevant as limits of integration too, because when dealing with absolute values you have to be careful about where things change signs. Fortunately, both of the functions have zeros at -1, and no other relevant zeros, so that's not an issue in this particular case.