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.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Volume by Shells Date Period Find the volume of each solid. Give exact values. The region under $y = 4 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 3 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 2 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 0$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 2 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 1$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 4 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 2 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 3 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 4 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 4 x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 3 x^{2}$ from $x = 2$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 2$ to $x = 3 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 2 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 3 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ The region under $y = 4 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ 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.

The region under $y = 4 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 3 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 2 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 0$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 2 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 1$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 2 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 3 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 4 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 3 x^{2}$ from $x = 2$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 2$ to $x = 3 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 2 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 3 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$
The region under $y = 4 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$



Sigma Prep · sigmaprep.io/worksheets Name Volume by ShellsAnswer keyVersion 1 Date Period Find the volume of each solid. Give exact values. The region under $y = 4 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $30 π$ The region under $y = 3 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $360 π$ The region under $y = 2 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $624 π$ The region under $y = x^{2}$ from $x = 0$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $8 π$ The region under $y = 2 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $1 π$ The region under $y = x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $312 π$ The region under $y = x^{2}$ from $x = 1$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{255}{2} π$ The region under $y = 4 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $480 π$ The region under $y = 2 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $256 π$ The region under $y = 3 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{45}{2} π$ The region under $y = x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{15}{2} π$ The region under $y = 4 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $1248 π$ The region under $y = 4 x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $2560 π$ The region under $y = x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $640 π$ The region under $y = 3 x^{2}$ from $x = 2$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{1827}{2} π$ The region under $y = x^{2}$ from $x = 2$ to $x = 3 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{65}{2} π$ The region under $y = 2 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $15 π$ The region under $y = x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $128 π$ The region under $y = 3 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{3}{2} π$ The region under $y = 4 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $512 π$ 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.

The region under $y = 4 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $30 π$
The region under $y = 3 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $360 π$
The region under $y = 2 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $624 π$
The region under $y = x^{2}$ from $x = 0$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $8 π$
The region under $y = 2 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $1 π$
The region under $y = x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $312 π$
The region under $y = x^{2}$ from $x = 1$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{255}{2} π$
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $480 π$
The region under $y = 2 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $256 π$
The region under $y = 3 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{45}{2} π$
The region under $y = x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{15}{2} π$
The region under $y = 4 x^{2}$ from $x = 1$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $1248 π$
The region under $y = 4 x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $2560 π$
The region under $y = x^{2}$ from $x = 2$ to $x = 6 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $640 π$
The region under $y = 3 x^{2}$ from $x = 2$ to $x = 5 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{1827}{2} π$
The region under $y = x^{2}$ from $x = 2$ to $x = 3 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{65}{2} π$
The region under $y = 2 x^{2}$ from $x = 1$ to $x = 2 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $15 π$
The region under $y = x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $128 π$
The region under $y = 3 x^{2}$ from $x = 0$ to $x = 1 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $\frac{3}{2} π$
The region under $y = 4 x^{2}$ from $x = 0$ to $x = 4 e x t i s s p u n a b o u t t h e y e x t - a x i s . F i n d t h e v o l u m e u s i n g s h e l l s .$ $512 π$

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.

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.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

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.

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

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Volume by Shells

Volume by ShellsAnswer keyVersion 1

How to do these

Where students go wrong

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