The area between curves is the size of the region enclosed by two or more graphs, calculated with definite integrals of (top function − bottom function) on each subinterval, or equivalently by integrating the absolute value of the difference |f(x) − g(x)| over the interval.
The area between curves is exactly what it sounds like, the amount of space trapped between two graphs. You find it by integrating the difference of the two functions, top minus bottom, over the interval where the region lives. The catch in Topic 8.6 is that curves can cross more than twice. Every time they intersect, the "top" and "bottom" functions can swap, so you can't just run one integral from start to finish with a fixed order.
The CED gives you two equivalent fixes. You can split the region at every intersection point and write a sum of definite integrals, flipping which function is on top in each piece. Or you can integrate the absolute value of the difference, ∫|f(x) − g(x)| dx, which automatically keeps every piece positive. Both moves come straight from essential knowledge under learning objective 8.6.A. If the curves are written as functions of y instead of x, the same logic applies sideways. You integrate (right − left) with respect to y.
This term lives in Unit 8: Applications of Integration, specifically Topic 8.6, and supports learning objective 8.6.A, which asks you to calculate areas in the plane using the definite integral. It's the payoff moment of the course. Everything you learned about antiderivatives and the Fundamental Theorem in Unit 6 finally measures something geometric. Area between curves is also the setup for almost everything else in Unit 8. Volumes by cross-sections and by revolution (Topics 8.7-8.12) all start with a region between curves, so if you can't define this region correctly, the rest of the unit collapses. On the AP exam, a region bounded by two curves is one of the most reliable FRQ openers in the entire course.
Keep studying AP Calculus Unit 8
Visual cheatsheet
view galleryPoints of Intersection (Unit 8)
Intersection points are the hinges of the whole problem. They become your limits of integration, and they mark exactly where top and bottom might swap. Solving f(x) = g(x), often on your calculator, is always step one.
Integration and the Fundamental Theorem (Unit 6)
Area between curves is the definite integral doing its day job. A plain definite integral gives net signed area relative to the x-axis; here you just replace "the x-axis" with a second curve and integrate the gap between them.
Functions of x and y (Unit 8)
Some regions are awkward with vertical slices but easy with horizontal ones. Rewriting the curves as functions of y and integrating (right − left) dy is the same idea rotated 90 degrees, and the AP exam expects you to choose the friendlier direction.
Velocity Function (Unit 8)
Total distance traveled is ∫|v(t)| dt, which is the same absolute-value trick. Distance is literally the area between the velocity curve and the t-axis, with sign flips handled by splitting at zeros, just like splitting at intersection points.
Area between curves shows up in both multiple choice and free response, and a shaded region bounded by two curves is a classic FRQ part (a), often calculator-active. Expect to do four things. First, find the intersection points by solving f(x) = g(x), frequently with a calculator since the curves rarely intersect at nice numbers. Second, figure out which function is on top in each subregion, since curves that cross more than twice force you to split the integral or use |f − g|. Third, set up the integral correctly, (top − bottom) dx or (right − left) dy, with the right limits. Fourth, evaluate. Setup errors are where points die. If you write one integral across an interval where the curves swap, the signed pieces cancel and your "area" comes out too small. Show your integral setup explicitly even on calculator problems, because the setup itself earns points.
A definite integral ∫f(x) dx gives net signed area, so parts below the x-axis count as negative and can cancel parts above it. Area between curves is always a positive geometric amount. That's why you integrate (top − bottom), splitting wherever the curves cross, or integrate |f − g|. If your answer for an area comes out negative or suspiciously small, you probably let the curves swap on you and computed a net value instead of a total.
Area between curves equals the integral of (top function minus bottom function), or (right minus left) if you slice horizontally with respect to y.
When curves intersect at more than two points, split the region at every intersection and write a sum of integrals, flipping top and bottom as needed.
Integrating the absolute value |f(x) − g(x)| over the whole interval is an equivalent shortcut that the CED explicitly allows under learning objective 8.6.A.
Intersection points come from solving f(x) = g(x), and they serve as your limits of integration, so find them first.
Area is always positive; if your integral gives a negative or unexpectedly small answer, the curves swapped somewhere and pieces canceled.
This skill is the foundation for volumes in Topics 8.7-8.12, so a wrong region setup ruins later parts of the same FRQ.
It's the size of the region enclosed by two or more graphs, found by integrating (top − bottom) over the interval, or by integrating |f(x) − g(x)|. It's the focus of Topic 8.6 in Unit 8 and supports learning objective 8.6.A.
No. Area is a positive geometric quantity. If your integral comes out negative, you subtracted in the wrong order (bottom minus top) or ran one integral across an intersection where the curves swapped, letting signed pieces cancel.
A regular definite integral ∫f(x) dx measures net signed area between one curve and the x-axis, so regions below the axis count negative. Area between curves measures total enclosed space between two graphs, which is why you use (top − bottom) on each piece or the absolute value of the difference.
Solve f(x) = g(x) to find every intersection, then split the integral at each one. On each subinterval, identify which curve is on top and integrate (top − bottom). Adding those integrals gives the total area, and integrating |f − g| does the same job in one step.
Use y when horizontal slices describe the region more cleanly, meaning when the curves are easier to write as functions of y or when vertical slices would hit the same curve twice. Then you integrate (right − left) dy between y-limits found from the intersection points.