Volumes of revolution

Integration adds up areas. Push the same idea one dimension further and it adds up volumes — starting with the solids you get by spinning a curve around an axis.

Spin a curve into a solid

Take the region under y = f(x) and revolve it around the x-axis. Every point sweeps out a circle, and the whole region sweeps out a solid of revolution. Slice that solid perpendicular to the axis and each slice is a thin disk.

The disk method

A disk at position x is a circle whose radius is the height of the curve, f(x). Its area is π·[f(x)]², and its thickness is dx, so its volume is π·[f(x)]²·dx. Add up the disks with an integral:

V = ∫ₐᵇ π [ f(x) ]² dx

It's the area-under-a-curve idea with the height replaced by a disk's area — you're integrating cross-sections instead of strip heights.

Worked example

Revolve y = √x from x = 0 to x = 4 around the x-axis.

• Radius = √x, so [f(x)]² = x.
• V = ∫₀⁴ π·x dx = π · [ x²/2 ]₀⁴ = π · (16/2 − 0) = 8π ≈ 25.1.

Squaring the radius is what makes [f(x)]² so clean here — the √ disappears.

A cross-section view of solids

This connects straight back to the geometry course: a cylinder is what you get by revolving a horizontal line (constant radius), and V = ∫ π r² dx = π r² · h — the familiar cylinder formula, now derived by integration rather than memorized. Cones and spheres fall out the same way from the right curve.

Why this matters

Disks are one case of a bigger idea: integrate the cross-sectional area and you get a volume, whatever the shape of the slices. It's how engineers find the volume of turned parts, bottles, lenses, and tanks — any solid with a describable profile.