Practice Volumes of Revolution in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

In the shell method about the

y

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

f(x)

at radius

x

Example 2

medium

Region under

y=x^2

from 0 to 2, revolved about the

y

-axis using shells.

Example 3

easy

Find the volume when

y=\sqrt{x}

0\le x\le 9

, is revolved about the

x

-axis.

Example 4

challenge

Region bounded by

y=x^2

y=0

x=2

revolved about the line

x=2

using shells.

Example 5

medium

The region under

y=1/x

1\le x\le 3

, is revolved about the

x

-axis. Find the volume.

Example 6

easy

What goes wrong with

V=\pi\int_0^1 f(x)\,dx

for a solid of revolution?

Example 7

easy

Region under

y=x

from 0 to 1 is revolved about the

x

-axis. Set up the disc-method integral.

Example 8

easy

Find the volume of the solid formed by rotating

f(x) = \sqrt{x}

around the

x

-axis from

x=0

x=4

Example 9

easy

Find the volume of the solid formed by rotating

y = 2x

from

x=0

x=3

around the

x

-axis.

Example 10

medium

Region under

y=\sqrt{x}

from 0 to 1 revolved about the

y

-axis (disc in

y

Example 11

medium

Find the volume when

y=x^3

0\le x\le 1

, is revolved about the

x

-axis.

Example 12

medium

Region under

y=x

from 0 to 2 revolved about the

y

-axis using shells.

Example 13

hard

The region between

y=x^2

and

y=x

0\le x\le 1

, is revolved about

y=1

. Find the volume.

Example 14

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 15

easy

Find the volume when

y=x

0\le x\le 2

, is revolved about the

x

-axis.

Example 16

easy

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

y

-axis but integrating in

x

Example 17

medium

Region under

y=x+1

from 0 to 2 revolved about the

x

-axis. Find the volume.

Example 18

medium

Region under

y=x^2

from 0 to 1 revolved about the

x

-axis. Find the volume.

Example 19

medium

The region bounded by

y=x

y=0

, and

x=1

is revolved about the line

y=-1

. Find the volume.

Example 20

medium

The region under

y=x^3

0\le x\le 1

, revolved about the

y

-axis using shells. Find the volume.

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Related Concepts

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