Area Between Curves

Overview

With Respect to x

With Respect to y

       

In an area between curves problem, you’ll be given two functions, and asked to find the area enclosed between the two of them. The steps you’ll need to follow in order to solve an area between curves problem are the following:

1. Set the functions equal to one another to find the points where they intersect
2. Between each of the points of intersection (and the end points of the range if they are given), determine which function has a higher value
3. Take the integral of the “higher function” minus the “lower function” for each range (this will be a definite integral), and add them all together

If the problem gives you a range on which you’re supposed to evaluate the area between the curves, you’ll need to find intersection points on that range. If the problem doesn’t give you a specific range, then it’s likely that you’re only being asked to evaluate the area between the curves between the points of intersection. Either way, to find the points of intersection, set the functions equal to one another and solve for . It’s possible to get no solutions, one solution, or multiple solutions.

If you get no solutions for , there are no points of intersection. Pick any value between the end points of your range and plug it into both functions. Whichever function gives you a greater value is the greater function. Because there are no points of intersection where the functions cross each other, this means that the greater function is greater on the entire range. Therefore, you’ll take the integral of the greater function minus the lesser function.

If you get one solution for when you set the functions equal to one another, then your range is essentially divided into two segments. Because the functions cross each other exactly once, then you know that one function is higher between the lower bound and the intersection point, and the other function is higher between the intersection point and the upper bound. You need to pick a value between the lower bound and the intersection point and plug it into both functions to see which function is higher on that segment of the graph. Then pick a value between the intersection point and the upper bound and plug it into both functions to see which function is higher on that segment of the graph. You’ll need to find the sum of two integrals. The first integral will represent the area between the lower bound and the intersection point. The second integral will represent the area between the intersection point and the upper bound. In both integrals, make sure to subtract the lower function from the upper function. The upper and lower functions will be opposite across the two integrals.

If you get multiple solutions for when you set the functions equal to one another, then your range is divided into multiple segments and you’ll need to test each one to determine on each range which function is higher than the other. As always, you’ll have a separate integral for each range, and in each one, you’ll subtract the lower function from the upper function.