A cross section is the two-dimensional shape formed where a plane cuts through a solid. On the AP Calculus exam (Topic 8.8), you find a solid's volume by writing a formula A(x) for the area of each cross section and integrating it across the base: V = ∫A(x)dx.
A cross section is the flat, 2D shape you see when you slice a solid object with a plane. Picture cutting a loaf of bread. Each slice is a cross section, and if you know the area of every slice, you can rebuild the volume of the whole loaf by adding the slices up. That "adding up infinitely thin slices" idea is exactly what a definite integral does.
In AP Calculus, cross-section problems give you a 2D base region (usually bounded by curves) and tell you that slices perpendicular to an axis form a known shape, like squares, equilateral triangles, or semicircles. Your job is to write the area of one slice as a function A(x), where the side or diameter of the shape comes from the distance between the curves. The CED spells this out in CHA-5.B.2 (triangular cross sections) and CHA-5.B.3 (semicircular and other geometrically defined cross sections). Then volume is V = ∫ from a to b of A(x) dx.
Cross sections live in Topic 8.8, Volumes with Cross Sections, in Unit 8 (Applications of Integration). Learning objective 8.8.A asks you to calculate volumes of solids with known cross sections using definite integrals, supported by CHA-5.B.2 and CHA-5.B.3. This topic is where geometry and integration fully merge. It also explains WHY the disk and washer methods in Topics 8.9-8.12 work, since a disk is just a circular cross section. If you understand cross sections, every volume formula in Unit 8 stops being a memorized template and becomes the same idea: integrate area to get volume.
Keep studying AP Calculus Unit 8
Visual cheatsheet
view galleryVolume (Unit 8)
Volume is what cross sections are for. The whole method is one sentence: volume equals the integral of cross-sectional area. The 2017 FRQ made this literal by handing you A(h), the area of the tank's horizontal cross section at height h, and asking you to work with ∫A(h)dh.
Base and Perpendicular Slices (Unit 8)
The base is the 2D region between curves, and cross sections sit perpendicular to an axis on top of it. The distance between the curves becomes the side length or diameter of each slice. Get the base wrong and every A(x) after it is wrong too.
Limits of Integration (Units 6 & 8)
The limits a and b mark where the base region starts and stops, often the intersection points of the bounding curves. On the 2022 FRQ, you had to find the intersection x = B with your calculator before you could even set up the volume integral.
Riemann Sums and the Definite Integral (Unit 6)
A cross-section volume is a Riemann sum in disguise. Each term A(x)Δx is a thin slab with face area A(x) and thickness Δx, and the integral is the limit as the slabs get infinitely thin. This is the cleanest example on the exam of why integrals accumulate quantities.
Cross sections show up two ways. First, the classic setup gives you a region R bounded by curves, then says something like "R is the base of a solid whose cross sections perpendicular to the x-axis are squares" (2024 FRQ Q6) or rectangles, triangles, or semicircles. You must write A(x) using the gap between the curves, set the limits of integration at the region's endpoints, and evaluate (often with a calculator on the AB exam). Second, the problem can skip the geometry and hand you the area function directly, like the 2017 tank FRQ where A(h) gave the horizontal cross-sectional area and you interpreted ∫A(h)dh as volume. Multiple-choice versions test setup more than evaluation, so watch for the area formula. A square slice uses (top − bottom)², a semicircle uses (1/2)π r² where the gap is the diameter, not the radius. That diameter-versus-radius slip is one of the most common point-losers in Unit 8.
Both are Unit 8 volume methods, but they answer to different setups. A revolution problem spins a region around an axis, and the cross sections automatically become circles (disks) or rings (washers). A general cross-section problem never rotates anything. The region just sits there as a base, and the problem tells you the slice shape, like squares or equilateral triangles. Quick test: if the prompt says "revolved about," use disks or washers; if it says "cross sections perpendicular to an axis are [shape]," use V = ∫A(x)dx with that shape's area formula.
A cross section is the 2D shape formed where a plane slices a solid, and integrating its area A(x) over the base gives the solid's volume: V = ∫A(x)dx.
This is learning objective 8.8.A in Unit 8, with the CED specifically naming triangular (CHA-5.B.2) and semicircular and other geometric cross sections (CHA-5.B.3).
The distance between the bounding curves of the base becomes the side, leg, or diameter of each cross section, so always write that gap first.
For semicircular cross sections, the gap between the curves is usually the diameter, so the radius is half of it before you plug into (1/2)πr².
The limits of integration come from where the base region begins and ends, often the intersection points of the curves.
Disks and washers are just the special case where cross sections are circles, so every volume method in Unit 8 is really the same integral of area.
It's the 2D shape you get when a plane slices through a solid. In Topic 8.8, you find volume by writing the area of each slice as A(x) and integrating it over the base region: V = ∫ from a to b of A(x) dx.
No. That's the key difference from disk and washer problems. In a cross-section problem the region is the base of the solid, nothing spins, and the prompt tells you the slice shape directly, like "cross sections perpendicular to the x-axis are squares."
The disk method is actually a cross-section problem where every slice happens to be a circle, because the solid was made by revolving a region around an axis. General cross-section problems (Topic 8.8) let the slices be any known shape, while disks and washers (Topics 8.9-8.12) only handle circles and rings.
Yes, regularly. The 2024 FRQ used square cross sections on a region between two parabolas, and the 2017 FRQ gave the cross-sectional area A(h) of a tank and asked you to interpret ∫A(h)dh as volume. Expect it in both MCQ and FRQ form on AB and BC.
V = ∫(1/2)πr² dx, where the radius r is half the distance between the curves, because that distance is the diameter. Forgetting to halve the diameter is one of the most common errors on this topic.