AP® Calculus AB · Unit 8 · Topic 8.6
Finding the Area Between Curves That Intersect at More Than Two Points
When two curves swap which one is on top more than once, you must split the interval at every crossing point and integrate top-minus-bottom separately on each piece, then add the results.
A single integral of (top minus bottom) only works if the same curve stays on top across the whole interval. If the curves cross in the middle, each one takes a turn being higher, so the expression f(x) - g(x) is positive on one piece and negative on the other. Integrating straight through adds a positive area to a negative one, and the two lobes partly cancel instead of combining.
The fix is to find every x-value where the curves meet by setting f(x) = g(x), then break the interval into pieces at those points. On each piece, check which function is actually on top — plug in a test value — and integrate that piece as (top minus bottom), which is always positive. Add up the absolute areas of all the pieces to get the total enclosed area.
This is the natural extension of the basic area-between-curves idea from earlier in the unit: same top-minus-bottom setup, just applied piece by piece instead of once. It also sets up solids of revolution and cross-sections later, where you will again need to know exactly where boundaries switch before setting up an integral.
Key terms
- intersection point
- An x-value where two curves have equal y-values, found by solving f(x) = g(x).
- top function
- On a given piece of the interval, whichever curve has the larger y-value there.
- signed area
- The result of a plain integral, which can be negative and causes cancellation when a function dips below another.
- lobe
- One enclosed piece of the region between two curves, bounded by consecutive intersection points.
Watch it explained
- How it worksHow do you split a region at each crossing point to keep every height positive?
- Worked exampleFind the area between y = x3 and y = x on the interval from -1 to 1.
- Common mistakeWhy isn't the integral of the difference the area once that difference changes sign?
- Ties togetherHow does splitting the interval at each crossing point extend a single top-minus-bottom area integral?
Check yourself
End of topic.