Volumes of Solids

You’ll be able to

Compute volumes by the disk and washer methods
Set up volumes of solids with known cross sections
Select the correct method from the geometry of the region

Slicing a solid into thin pieces

To find a volume, slice the solid into thin slabs perpendicular to an axis, find the area A(x) of a typical cross section, and integrate: V = ∫ₐᵇ A(x) dx. The whole subject is just choosing the right cross-sectional area. When a region is revolved around an axis, the slices are disks (area πr²) or washers (area π(R² − r²)); for a solid with a stated cross-section shape, A(x) is that shape’s area built from the region’s height.

Disk and washer methods

disk: V = π ∫ₐᵇ [R(x)]² dx · washer: V = π ∫ₐᵇ ([R(x)]² − [r(x)]²) dx

R is the outer radius (to the far edge), r the inner radius (to the hole). Use a washer whenever the region does not touch the axis.

Known cross sections

Not every solid comes from revolution. A solid may sit on a base region in the plane with cross sections perpendicular to an axis shaped as squares, semicircles, or equilateral triangles. Here the region’s height at x gives the side (or diameter), you plug it into the shape’s area formula to get A(x), and integrate. For a square cross section A(x) = [side]²; for a semicircle A(x) = (π/8)(diameter)²; for an equilateral triangle A(x) = (√3/4)(side)².

Worked example

The region under y = √x from x = 0 to x = 4 is revolved about the x-axis. Find the volume.

1.Revolving about the x-axis with the region touching the axis gives solid disks of radius R(x) = √x.
2.Disk area: A(x) = π[R(x)]² = π(√x)² = πx.
3.Set up the volume integral: V = ∫₀⁴ πx dx.
4.Evaluate: π[x²/2]₀⁴ = π(16/2) = 8π.

Answer: The volume is 8π. Squaring the radius √x conveniently gives x, making the integrand simply πx.

Watch out

Use a washer, not a disk, whenever there is a gap between the region and the axis of revolution. Forgetting the inner radius r and treating a washer as a disk overcounts the volume by filling in the hole.

Checkpoint

A region is revolved about the x-axis and the region touches the axis (no gap). Which method applies, and what is the cross-sectional area?

Checkpoint

A solid has a base bounded by curves, and its cross sections perpendicular to the x-axis are squares with side length s(x). The volume integral is:

On the exam

For cross-section problems, first decide whether the region’s height gives the side or the diameter of the shape. Semicircle and some square-on-the-diagonal setups use the height as a diameter, which changes the area formula — misreading this is a common lost point.

Answer the 2 checkpoints as you read.