Volume by cylindrical shells worksheet

Spin a region about the y axis and a thin strip at x sweeps out a tube. Unroll the tube and it is a rectangle: its height times its circumference, which is where the 2 pi x comes from.

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Sigma Prep · sigmaprep.io/worksheets Name Volume by Shells Date Period Find the volume of each solid. Give exact values. Find volumes by cylindrical shells The region under $y = 5 - x$ from $x = 0$ to $x = 5$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. Use shells to find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 1$ is revolved about the $y$ -axis. The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 1$ and $x = 2$ is spun about the $y$ -axis. Find the volume using shells. Use shells to find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 5$ is revolved about the $y$ -axis. The region between $y = x$ and $y = x^{2}$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. Use shells to find the volume when the region bounded by $y = x^{2}$ , $y = 0$ , and $x = 1$ is revolved about the line $x = 2$ . The region bounded by $y = 5 - x^{2}$ , the axes and $x = 2$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. Use shells to find the volume when the region bounded by $y = 9 - x^{2}$ , $y = 0$ , and $x = 0$ is revolved about the $y$ -axis. The region under $y = 4 x^{2}$ from $x = 2$ to $x = 5$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. Use shells to find the volume when the region bounded by $y = x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. The region between $y = 3 x$ and $y = x^{2}$ is spun about the $y$ -axis. Find the volume using shells. Use shells to find the volume when the region bounded by $y = 2 x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 6$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. Use shells about a vertical line The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = 5$ . Use shells to find the volume. Use cylindrical shells to find the volume when the region between $y = 2 x - x^{2}$ and the $x$ -axis is revolved about the line $x = - 8$ . Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = - 7$ , using shells. By the shell method, what volume does the region under $y = x^{3}$ on $[0, 1]$ sweep out when revolved about $x = 1$ ? Spin the region under $y = x^{3}$ on $[0, 1]$ about the vertical line $x = - 3$ . Set up and evaluate the shell integral for the volume. Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = 2$ , using shells. The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = - 1$ . Use shells to find the volume. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Volume by Shells

Date Period

Find the volume of each solid. Give exact values.

Find volumes by cylindrical shells
The region under $y = 5 - x$ from $x = 0$ to $x = 5$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid.
Use shells to find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 1$ is revolved about the $y$ -axis.
The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 1$ and $x = 2$ is spun about the $y$ -axis. Find the volume using shells.
Use shells to find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 5$ is revolved about the $y$ -axis.
The region between $y = x$ and $y = x^{2}$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid.
Use shells to find the volume when the region bounded by $y = x^{2}$ , $y = 0$ , and $x = 1$ is revolved about the line $x = 2$ .
The region bounded by $y = 5 - x^{2}$ , the axes and $x = 2$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid.
Use shells to find the volume when the region bounded by $y = 9 - x^{2}$ , $y = 0$ , and $x = 0$ is revolved about the $y$ -axis.
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 5$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method.
Use shells to find the volume when the region bounded by $y = x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis.
The region between $y = 3 x$ and $y = x^{2}$ is spun about the $y$ -axis. Find the volume using shells.
Use shells to find the volume when the region bounded by $y = 2 x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis.
The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 6$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method.

Use shells about a vertical line
The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = 5$ . Use shells to find the volume.
Use cylindrical shells to find the volume when the region between $y = 2 x - x^{2}$ and the $x$ -axis is revolved about the line $x = - 8$ .
Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = - 7$ , using shells.
By the shell method, what volume does the region under $y = x^{3}$ on $[0, 1]$ sweep out when revolved about $x = 1$ ?
Spin the region under $y = x^{3}$ on $[0, 1]$ about the vertical line $x = - 3$ . Set up and evaluate the shell integral for the volume.
Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = 2$ , using shells.
The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = - 1$ . Use shells to find the volume.



Sigma Prep · sigmaprep.io/worksheets Name Volume by ShellsAnswer keyVersion 1 Date Period Find the volume of each solid. Give exact values. Find volumes by cylindrical shells The region under $y = 5 - x$ from $x = 0$ to $x = 5$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $\frac{125 π}{3}$ Use shells to find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 1$ is revolved about the $y$ -axis. $\frac{4 π}{5}$ The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 1$ and $x = 2$ is spun about the $y$ -axis. Find the volume using shells. $30 π$ Use shells to find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 5$ is revolved about the $y$ -axis. $8 π$ The region between $y = x$ and $y = x^{2}$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $\frac{π}{6}$ Use shells to find the volume when the region bounded by $y = x^{2}$ , $y = 0$ , and $x = 1$ is revolved about the line $x = 2$ . $\frac{5 π}{6}$ The region bounded by $y = 5 - x^{2}$ , the axes and $x = 2$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $12 π$ Use shells to find the volume when the region bounded by $y = 9 - x^{2}$ , $y = 0$ , and $x = 0$ is revolved about the $y$ -axis. $\frac{81 π}{2}$ The region under $y = 4 x^{2}$ from $x = 2$ to $x = 5$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. $1218 π$ Use shells to find the volume when the region bounded by $y = x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. $\frac{π}{162}$ The region between $y = 3 x$ and $y = x^{2}$ is spun about the $y$ -axis. Find the volume using shells. $\frac{27 π}{2}$ Use shells to find the volume when the region bounded by $y = 2 x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. $\frac{8 π}{81}$ The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 6$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. $2560 π$ Use shells about a vertical line The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = 5$ . Use shells to find the volume. $\frac{71 π}{6}$ Use cylindrical shells to find the volume when the region between $y = 2 x - x^{2}$ and the $x$ -axis is revolved about the line $x = - 8$ . $24 π$ Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = - 7$ , using shells. $\frac{113 π}{30}$ By the shell method, what volume does the region under $y = x^{3}$ on $[0, 1]$ sweep out when revolved about $x = 1$ ? $\frac{π}{10}$ Spin the region under $y = x^{3}$ on $[0, 1]$ about the vertical line $x = - 3$ . Set up and evaluate the shell integral for the volume. $\frac{19 π}{10}$ Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = 2$ , using shells. $\frac{11 π}{15}$ The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = - 1$ . Use shells to find the volume. $\frac{25 π}{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Volume by ShellsAnswer keyVersion 1

Date Period

Find the volume of each solid. Give exact values.

Find volumes by cylindrical shells
The region under $y = 5 - x$ from $x = 0$ to $x = 5$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $\frac{125 π}{3}$
Use shells to find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 1$ is revolved about the $y$ -axis. $\frac{4 π}{5}$
The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 1$ and $x = 2$ is spun about the $y$ -axis. Find the volume using shells. $30 π$
Use shells to find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 5$ is revolved about the $y$ -axis. $8 π$
The region between $y = x$ and $y = x^{2}$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $\frac{π}{6}$
Use shells to find the volume when the region bounded by $y = x^{2}$ , $y = 0$ , and $x = 1$ is revolved about the line $x = 2$ . $\frac{5 π}{6}$
The region bounded by $y = 5 - x^{2}$ , the axes and $x = 2$ is revolved around the $y$ -axis. Use cylindrical shells to find the volume of the solid. $12 π$
Use shells to find the volume when the region bounded by $y = 9 - x^{2}$ , $y = 0$ , and $x = 0$ is revolved about the $y$ -axis. $\frac{81 π}{2}$
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 5$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. $1218 π$
Use shells to find the volume when the region bounded by $y = x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. $\frac{π}{162}$
The region between $y = 3 x$ and $y = x^{2}$ is spun about the $y$ -axis. Find the volume using shells. $\frac{27 π}{2}$
Use shells to find the volume when the region bounded by $y = 2 x$ and $y = 3 x^{2}$ is revolved about the $y$ -axis. $\frac{8 π}{81}$
The region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 6$ is rotated about the $y$ -axis. Find the volume of the solid by the shell method. $2560 π$

Use shells about a vertical line
The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = 5$ . Use shells to find the volume. $\frac{71 π}{6}$
Use cylindrical shells to find the volume when the region between $y = 2 x - x^{2}$ and the $x$ -axis is revolved about the line $x = - 8$ . $24 π$
Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = - 7$ , using shells. $\frac{113 π}{30}$
By the shell method, what volume does the region under $y = x^{3}$ on $[0, 1]$ sweep out when revolved about $x = 1$ ? $\frac{π}{10}$
Spin the region under $y = x^{3}$ on $[0, 1]$ about the vertical line $x = - 3$ . Set up and evaluate the shell integral for the volume. $\frac{19 π}{10}$
Find the volume of the solid made by revolving the region in the first quadrant between $y = x$ and $y = x^{3}$ about $x = 2$ , using shells. $\frac{11 π}{15}$
The region under $y = x^{2} + 1$ on $[0, 1]$ is revolved about $x = - 1$ . Use shells to find the volume. $\frac{25 π}{6}$

How to do these

The region under $y = x$ from $x = 0$ to $x = 2$ is spun about the $y$ -axis.

A strip at x sweeps out a tube of radius x and height x.
So the volume is 2 pi times the integral of x times x, from 0 to 2.
That is 2 pi times 8 over 3, which is 16 pi over 3.

The answer is $\frac{16}{3} π$ .

Where students go wrong

Forgetting the x in front. The 2 pi x is the circumference of the tube, and without it the integral is the area of the region rather than the volume of the solid.

Also called shell method, cylindrical shells or volume about the y-axis.

What you can put on this worksheet

Volume by cylindrical shells: The region bounded by $y = 6$ , the $x$ -axis, $x = 1$ and $x = 4$ is spun about the $y$ -axis. Find the volume using shells. → $90 π$
About other vertical lines: The region under $y = 2 x$ on $[0, 1]$ is revolved about $x = - 7$ . Use shells to find the volume. → $\frac{46 π}{3}$
Two curves, roots, and about x = -c: Use shells to find the volume when the region bounded by $y = x^{3}$ , $y = 0$ , and $x = 2$ is revolved about the $y$ -axis. → $\frac{64 π}{5}$

Questions about these worksheets

Yes. Every volume by cylindrical shells sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

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Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the volume by cylindrical shells questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: volume by cylindrical shells, about other vertical lines and two curves, roots, and about x = -c. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the x in front. The 2 pi x is the circumference of the tube, and without it the integral is the area of the region rather than the volume of the solid.

Volume by cylindrical shells worksheet

Select your difficulty

Find volumes by cylindrical shells

Use shells about a vertical line

Find volumes by cylindrical shells

Use shells about a vertical line

How to do these

Where students go wrong

What you can put on this worksheet

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