8.6 Finding the Area Between Curves That Intersect at More Than Two Points

2 min read•june 18, 2024

AP Calculus AB/BC ♾️

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When Curves Intersect More Than Once

This topic is very similar to the previous two topics. However, this topic involves more steps. Normally, you’d integrate the function (top minus bottom) or (right minus left) from the endpoints provided. In the situation where the curves intersect at more than two points, you must use additional steps:

Graph the functions
Identify the areas and what approach to take (either top minus bottom with vertical slices or right minus left with horizontal slices)
Set the two equations equal to each other and find the intersection points
Integrate from the intersection points found in step 3 to find the area

If you’re still confused, take a look at this example:

🔍 Example Problem: Finding the Area Between Two Curves That Intersect at More Than Two Points

Let’s say we want to find the area between y = sin(x) and y = -sin(x) from 0 to 2π.

First, we graph the functions. From this, we find that the functions intersect at more than the endpoints.
We have two areas, and the curves are on the top and bottom for both. Thus, we will use vertical slices to find the areas.
Third, we set the functions equal to each other to find the intersection point. Now, we know the intersection point is x = π.
Fourth, we integrate (sin(x) - -sin(x)) from 0 to π to find the first area. Lastly, we integrate (-sin(x) - sin(x)) from π to 2π to find the second area.
Add them together to get the final area.