AP® Calculus BC · Unit 8 · Topic 8.6
Area Between Curves With More Than Two Intersections
When two curves cross more than twice, which one is on top switches at each crossing, so you must split the interval at every intersection and integrate each piece separately before adding the areas.
Area between curves always means top minus bottom, but top and bottom can trade places. Every time the curves cross, they swap order: whoever was winning starts losing. If you integrate one fixed difference, like f(x) - g(x), across a stretch where that swap happens, the integral goes negative on the part where g is actually on top. That negative chunk subtracts from the positive chunk instead of adding to it, so your total comes out too small. It is still a correct number for something — the signed, net area — just not the actual area of the region you can see.
The fix is mechanical. Set f(x) = g(x), solve for every solution in the interval, and use those x-values as breakpoints. On each piece between consecutive breakpoints, one curve stays on top the whole time, so you integrate the absolute difference, or whichever order is correct for that piece, and get a positive number. Add up all the pieces. This is the same idea as area under a curve that dips below the x-axis: you split at the zeros and integrate each part separately so the areas add instead of cancel.
This topic is really unit 7's area-between-curves idea plus unit 6's instinct for splitting integrals at sign changes, now applied to three or more crossings instead of two. It sets up the same splitting habit used later for volumes of solids whose cross-sections change which curve bounds them, so getting comfortable finding every intersection point and checking order on each piece pays off again soon.
Key terms
- intersection point
- An x-value where the two curves have equal y-values, found by solving f(x) = g(x).
- signed area
- The result of integrating one fixed difference without splitting, where regions below zero subtract instead of add.
- total area
- The actual geometric area enclosed by the curves, found by splitting at each crossing and summing positive pieces.
- breakpoint
- An intersection x-value used to divide the interval into pieces where one curve stays on top throughout.
Watch it explained
- How it worksHow do I locate every intersection and split the region where the curves swap order?
- Worked exampleFind the area between y = x3 and y = x on the interval from -1 to 1.
- Common mistakeWhy doesn't one single integral of the difference work when the curves cross in the middle?
- Going furtherWhat happens to the total when the signs are allowed to cancel each other out?
Check yourself
End of topic.