AP® Calculus AB · Unit 8 · Topic 8.7
Volumes with Cross Sections: Squares and Rectangles
A solid can be built by stacking flat slices along an axis instead of spinning a region. If you know the area of each slice as a function of x, the volume is just the integral of that area function.
Picture a region in the xy-plane acting as a footprint. At every x-value, you erect a flat shape, say a square, standing straight up off the plane, perpendicular to the x-axis. No rotation happens. You're stacking infinitely many thin slabs side by side, each one a tiny prism with area A(x) and thickness dx. Add up all those slab volumes with an integral and you get the volume of the whole solid: V = integral from a to b of A(x) dx.
The mechanism hinges on finding A(x) correctly. For square cross sections, the side length of each square is the vertical distance between the two curves bounding the region at that x, so A(x) = (top - bottom)2. For a base between y = x2 and y = 4, the side length at each x is 4 - x2, so V = integral from -2 to 2 of (4 - x2)2 dx. You can't just multiply the base's flat area by a height, because the cross sections change size as x changes. Height times base area only works for prisms with constant cross section; here the square shrinks and grows, so you need the integral to track that varying area.
Rectangles twice as tall as wide work the same way, but the area formula changes: if the width is still (top - bottom), the height is 2(top - bottom), so A(x) = 2(top - bottom)2, exactly double the square case. This connects directly to the disc method from earlier in the unit: a disc is nothing but a cross section shaped like a circle, with A(x) = π r2 instead of a square's side2. Same idea, same integral, different shape.
Key terms
- Cross section
- A flat slice of a solid made by cutting perpendicular to a chosen axis at a specific point.
- Cross-sectional area function A(x)
- A formula giving the area of the slice at each x-value, used as the integrand for volume.
- Side length
- For square or rectangular cross sections, the length of one edge, often equal to the distance between two bounding curves.
- Disc method
- A special case of cross-sectional volume where each slice is a circle with area π r2.
Watch it explained
- How it worksHow does a cross-sectional area function integrate into a volume?
- Worked exampleFind the volume of a solid whose base lies between y = x2 and y = 4 with square cross sections.
- Common mistakeWhy isn't the volume just the base area multiplied by the height?
- Going furtherWhat changes when the squares become rectangles twice as tall as they are wide?
- Ties togetherHow is a disc in the disc method just a cross section that happens to be a circle?
Check yourself
End of topic.