Cross-Sections Versus Solids of Revolution

Two Different Ways to Specify a Solid

A solid can be described by what its slices look like, or by how a planar region moves when it is rotated. Both descriptions lead to volume integrals, but they provide different geometric information. In one case, the slice shape is prescribed directly; in the other, the rotation determines the slice shape and its dimensions.

This distinction matters when setting up an integral. The base region alone does not determine a solid made from prescribed cross-sections: the cross-sectional shape must also be given. For a solid of revolution, the region and axis of rotation determine the slices created by the rotation.

Solids Specified by Cross-Sections

Suppose a planar region serves as the base of a solid. At each position along an interval, a plane perpendicular to a chosen direction cuts the solid in a specified shape. The base gives a length or other dimension of that shape, and the problem tells you how the cross-section is constructed from that information.

For example, a base segment might be the side of a square cross-section, the diameter of a semicircle, or the base of a triangle. The corresponding areas are not interchangeable. If the relevant length at position \(x\) is \(s(x)\), then a square has area \(s(x)^2\); a semicircle with diameter \(s(x)\) has area \(\frac{\pi}{8}s(x)^2\). The area function must reflect the actual prescribed shape.

Solids Created by Rotation

A solid of revolution is formed by rotating a planar region about a line in its plane, called the axis of rotation. The rotation itself determines the cross-section: when each slice of the region is a single interval, a slice perpendicular to the axis is circular (a disk or a washer); a region with several intervals in one slice produces a union of such pieces. If the region reaches the axis, the slice is a disk. If the region lies between two boundaries at different distances from the axis, the slice is a washer.

The area is therefore determined by the radius or radii measured from the axis—not by treating an arbitrary boundary length as a side of some other shape. For a disk of radius \(R\), the area is \(\pi R^2\). For a washer with outer radius \(R\) and inner radius \(r\), where \(0\le r\le R\), the area is \(\pi(R^2-r^2)\). These are the circular-slice formulas developed in the earlier material on disks and washers.

What Determines the Integral?

In either situation, the essential quantity is the area of a slice at each position. If the slices are perpendicular to the \(x\)-axis and extend from \(x=a\) to \(x=b\), write \(A(x)\) for the cross-sectional area. The volume is obtained by integrating that area over the interval, provided the solid and area function satisfy the regularity conditions for the slicing principle.

For a prescribed cross-section, \(A(x)\) comes from the specified shape and the dimensions supplied by the base. For a solid of revolution, \(A(x)\) comes from a disk or washer determined by the radii from the axis. In both cases, the variable and interval must match the direction and extent of the slices.

Why the Slicing Principle Works

The basic idea is to approximate the solid by thin slabs. A slab with thickness \(\Delta t\) near position \(t\) has volume approximately \(A(t)\Delta t\). Adding these approximate slab volumes gives a Riemann sum for the integral of the cross-sectional area. As the slab widths approach zero, the sums approach the volume when the solid is regular enough for this approximation to apply.

There is also a precise formulation for bounded measurable solids. Use coordinates \((t,u,v)\), with \(t\) along the direction of slicing and \((u,v)\) in the perpendicular plane. Let \(\mathbf{1}_S(t,u,v)\) equal \(1\) when \((t,u,v)\) belongs to the solid \(S\), and \(0\) otherwise. The volume is the integral of this indicator over space. For a bounded measurable solid, Fubini’s theorem allows that integral to be evaluated by first integrating over each perpendicular plane and then over \(t\): the section areas \(A(t)\) are defined for almost every \(t\), and the inner integral is exactly \(A(t)\). Thus $$ \operatorname{Vol}(S) =\int_a^b\left(\iint_{\text{slice plane}}\mathbf{1}_S(t,u,v)\,du\,dv\right)dt =\int_a^b A(t)\,dt. $$ The measurability hypothesis makes this use of Fubini valid; it cannot be dropped merely because the area function happens to be integrable.

Worked Example: Square Cross-Sections

Worked Example: Semicircular Cross-Sections

Worked Example: Rotating the Same Base

A Common Setup Mistake

A frequent error is to use a length as though it were already an area. In the square example, the side at \(x\) is \(3-x\). Integrating that side directly would give

This is not the volume of the square-cross-section solid. The integrand \(3-x\) is a side length, not the area of a square. The correction is to square the side first: \(A(x)=(3-x)^2\), so the volume is \(\int_0^3(3-x)^2\,dx=9\). For a rotation about the \(x\)-axis, the same length is a radius, so the area instead is \(\pi(3-x)^2\).

Before integrating, identify what the given dimension represents: a side, diameter, radius, or distance between boundaries. Then use the formula for the specified cross-section. A boundary length does not determine the area until the shape and its relevant dimensions are known.

Why the Distinction Matters

These methods answer different geometric questions. A prescribed-cross-section problem tells you which shapes to stack and how their dimensions relate to the base. A solid-of-revolution problem tells you how a region sweeps around an axis; its circular slices follow from that motion. The shared integral formula does not make the solids or their area functions the same.

A reliable setup can be organized as follows:

The decisive step is constructing the correct area function. Once the slice geometry is correctly represented, the volume integral records the accumulated areas across the solid.