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.

Read the full concept explanation →

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: Volumes of revolution rotate a 2D region around an axis and integrate the areas of disc, washer, or shell cross-sections.

Common stuck point: The procedure for volumes of revolution is the easy part; the trap is forgetting the $\pi$ in disc/washer formulas. Asking "Is a flat region being spun around an axis to make a solid whose volume I find by integrating cross-sections?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is a flat region being spun around an axis to make a solid whose volume I find by integrating cross-sections?

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

x=4

Answer

V = 8\pi

First step

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

Full solution

2
Set up the integral: $V = \pi\int_0^4 x\,dx$
3
Evaluate: $V = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot\frac{16}{2} = 8\pi$

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.

Example 3

medium

Find the volume when

y=x^3

0\le x\le 1

, is revolved about the

x

-axis.

Example 4

medium

Use the washer method to find the volume when the region between

y=\sqrt{x}

(outer) and

y=x

(inner),

0\le x\le 1

, is revolved about the

x

-axis.

Example 5

medium

The region under

y=\sqrt{x}

0\le x\le 4

, is revolved about the

y

-axis using shells. Find the volume.

Example 6

hard

The region bounded by

y=x^2

x=2

y=0

is revolved about the line

x=2

using shells. Find the volume.

Example 7

hard

The region under

y=\sqrt{4-x^2}

-2\le x\le 2

, revolved about the

x

-axis. Identify the solid and find its volume.

Example 8

challenge

The region under

y=\sqrt{x}

0\le x\le 1

, revolved about the line

x=-1

using shells. Find the volume.

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

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.

Example 3

easy

Region under

y=x

from 0 to 1 is revolved about the

x

-axis. Set up the disc-method integral.

Example 4

easy

Evaluate the volume:

y=x

from 0 to 1 revolved about the

x

-axis.

Example 5

easy

Region under

y=\sqrt{x}

from 0 to 4 is revolved about the

x

-axis. Find the volume.

Example 6

easy

In the washer method, what is the cross-sectional area with outer radius

R

and inner radius

r

Example 7

easy

Region under

y=2

from

x=0

x=3

revolved about the

x

-axis. Find the volume.

Example 8

easy

If a region is revolved about

y=-1

, what is the radius for a curve

y=f(x)

above it?

Example 9

easy

In the shell method about the

y

-axis, what is one shell's volume for height

f(x)

at radius

x

Example 10

easy

Which method (disc/washer or shell) is natural when rotating about the

y

-axis but integrating in

x

Example 11

medium

Region under

y=x^2

from 0 to 1 revolved about the

x

-axis. Find the volume.

Example 12

medium

Region between

y=x

(outer) and

y=x^2

(inner) from 0 to 1 revolved about the

x

-axis.

Example 13

medium

Region under

y=x^2

from 0 to 2, revolved about the

y

-axis using shells.

Example 14

medium

Region under

y=\sqrt{x}

from 0 to 1 revolved about the

y

-axis (disc in

y

Example 15

medium

The region bounded by

y=x

y=0

, and

x=1

is revolved about the line

y=-1

. Find the volume.

Example 16

medium

Region under

y=x

from 0 to 2 revolved about the

y

-axis using shells.

Example 17

challenge

Region between

y=x

and

y=x^2

revolved about the

x

-axis; find the volume.

Example 18

challenge

Region under

y=x^2

from 0 to 2 revolved about the

x

-axis; find the volume.

Example 19

challenge

Region bounded by

y=x^2

y=0

x=2

revolved about the line

x=2

using shells.

Example 20

medium

Region under

y=x+1

from 0 to 2 revolved about the

x

-axis. Find the volume.

Example 21

medium

Region under

y=x^2

from 0 to 1 revolved about the

y

-axis using shells. Find the volume.

Example 22

medium

Region between

y=2

(outer) and

y=x

(inner) from 0 to 2 revolved about the

x

-axis.

Example 23

easy

Find the volume when

y=3

from

x=0

x=5

is revolved about the

x

-axis.

Example 24

easy

Find the volume when

y=x

0\le x\le 2

, is revolved about the

x

-axis.

Example 25

easy

Find the volume when

y=\sqrt{x}

0\le x\le 9

, is revolved about the

x

-axis.

Example 26

easy

Set up (do not evaluate) the disc integral for

y=x^3

0\le x\le 1

, revolved about the

x

-axis.

Example 27

medium

Find the volume when the region between

y=4-x^2

and the

x

-axis,

-2\le x\le 2

, is revolved about the

x

-axis.

Example 28

medium

The region under

y=1/x

1\le x\le 3

, is revolved about the

x

-axis. Find the volume.

Example 29

medium

The region under

y=x

0\le x\le 3

, is revolved about the

y

-axis using shells. Find the volume.

Example 30

medium

The region bounded by

y=x^2

and

y=4

is revolved about the

x

-axis. Find the volume.

Example 31

medium

The region under

y=e^x

0\le x\le 1

, is revolved about the

x

-axis. Find the volume.

Example 32

medium

The region under

y=x^2

above

y=0

0\le x\le 1

, is revolved about

y=-1

. Find the volume.

Example 33

medium

The region under

y=x^3

0\le x\le 1

, revolved about the

y

-axis using shells. Find the volume.

Example 34

medium

The region between

y=2x

and

y=x^2

0\le x\le 2

, is revolved about the

x

-axis. Find the volume.

Example 35

hard

The region under

y=\ln x

1\le x\le e

, revolved about the

y

-axis using shells. Find the volume.

Example 36

hard

The region under

y=x

0\le x\le 3

, revolved about the line

x=4

using shells. Find the volume.

Example 37

hard

The region between

y=x^2

and

y=x

0\le x\le 1

, is revolved about

y=1

. Find the volume.

← Back to Volumes of Revolution Formula Explained Practice Mode

Related Concepts

Area Between Curves Definite Integral Arc Length Surface Area of a Cylinder

Background Knowledge

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

area between curvesdefinite integral