Volumes of Revolution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Volumes of Revolution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.

Spin a flat region around a line, like spinning a pottery wheel. The flat shape sweeps out a 3D solid. To find its volume, slice the solid into thin pieces (discs, washers, or shells), find the volume of each slice, and add them up—which means integrate.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The volume of revolution is computed by integrating cross-sectional areas. The choice between disc/washer and shell methods depends on which makes the integral simpler—disc/washer integrates perpendicular to the axis of rotation, while shells integrate parallel to it.

Common stuck point: Choosing between disc/washer and shell methods: if the axis of rotation is horizontal and the function is given as y = f(x), disc/washer (integrating with dx) is often natural. If rotating around the y-axis, shells (integrating with dx) may be simpler than converting to x = g(y).

Sense of Study hint: Sketch the 2D region and the axis of rotation, then draw one representative slice (disc, washer, or shell) and label its radius.

Worked Examples

Example 1

easy
Find the volume of the solid formed by rotating f(x) = \sqrt{x} around the x-axis from x=0 to x=4.

Solution

  1. 1
    Use the disc method for revolution about the x-axis: V = \pi\int_a^b [f(x)]^2\,dx. Since f(x) = \sqrt{x}, [f(x)]^2 = x.
  2. 2
    Set up the integral: V = \pi\int_0^4 x\,dx
  3. 3
    Evaluate: V = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot\frac{16}{2} = 8\pi

Answer

V = 8\pi
The disc method stacks thin circular discs. Each has radius f(x) and thickness dx, giving volume element \pi[f(x)]^2\,dx.

Example 2

hard
Find the volume when the region between y=x and y=x^2 (0 \leq x \leq 1) is rotated around the x-axis.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the volume of the solid formed by rotating y = 2x from x=0 to x=3 around the x-axis.

Example 2

medium
Use the shell method to find the volume when the region bounded by y = x^2, x=0, y=1 is rotated around the y-axis.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

area between curvesdefinite integral