Volume by disks and washers worksheet

Spin a region about an axis and every slice through it is a circle, or a washer when the region does not touch the axis. Adding up those areas is an integral: pi times the radius squared, or pi times outer squared minus inner squared, all the way along.

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Sigma Prep · sigmaprep.io/worksheets Name Volume by Disks and Washers Date Period Find the volume of each solid. Give exact values. Find volumes by disks and washers Find the volume of the solid formed when the region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 3$ is revolved about the $x$ -axis. Find the volume when the region between $y = x$ and the $y$ -axis, for $0 \leq y \leq 5$ , is revolved about the $y$ -axis. The region bounded by $y = 3 x^{2}$ , the $y$ -axis and $y = 8$ is spun about the $y$ -axis. Find the volume. The region under $y = \sqrt{4 x}$ from $x = 1$ to $x = 4$ is spun about the $x$ -axis. Find the volume. Find the volume when the region between $y = x^{2}$ and the $y$ -axis, for $0 \leq y \leq 2$ , is revolved about the $y$ -axis. The region bounded by $y = 4 x$ , the $y$ -axis and $y = 9$ is spun about the $y$ -axis. Find the volume. Use disks to find the volume of the solid made by rotating the region under $y = 4 x^{2}$ from $x = 1$ to $x = 2$ about the $x$ -axis. Find volumes of revolution The region bounded by $y = x^{2}$ , $y = 1$ and the $y$ -axis in the first quadrant is spun about the line $x = 3$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 0$ and $x = 1$ is spun about the line $x = 1$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 2$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 4$ and the $y$ -axis in the first quadrant is spun about the line $x = 4$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 0$ and $x = 3$ is spun about the line $x = 3$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 1$ . Find the volume. The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = 2$ . Find the volume. Find volumes by revolving a region Find the volume when the region bounded by $x = y^{2}$ and $x = 4$ is revolved about the line $x = 4$ . Find the volume when the region bounded by $y = 2 x$ and $y = 2 x^{2}$ is revolved about the $x$ -axis. Find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 6$ is revolved about the $x$ -axis. Find the volume when the region bounded by $x = y^{2}$ and $x = 16$ is revolved about the $y$ -axis. Find the volume when the region bounded by $y = \sqrt{x}$ and $y = x$ is revolved about the line $y = - 3$ . Find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 6$ is revolved about the $x$ -axis. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Volume by Disks and Washers

Date Period

Find the volume of each solid. Give exact values.

Find volumes by disks and washers
Find the volume of the solid formed when the region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 3$ is revolved about the $x$ -axis.
Find the volume when the region between $y = x$ and the $y$ -axis, for $0 \leq y \leq 5$ , is revolved about the $y$ -axis.
The region bounded by $y = 3 x^{2}$ , the $y$ -axis and $y = 8$ is spun about the $y$ -axis. Find the volume.
The region under $y = \sqrt{4 x}$ from $x = 1$ to $x = 4$ is spun about the $x$ -axis. Find the volume.
Find the volume when the region between $y = x^{2}$ and the $y$ -axis, for $0 \leq y \leq 2$ , is revolved about the $y$ -axis.
The region bounded by $y = 4 x$ , the $y$ -axis and $y = 9$ is spun about the $y$ -axis. Find the volume.
Use disks to find the volume of the solid made by rotating the region under $y = 4 x^{2}$ from $x = 1$ to $x = 2$ about the $x$ -axis.

Find volumes of revolution
The region bounded by $y = x^{2}$ , $y = 1$ and the $y$ -axis in the first quadrant is spun about the line $x = 3$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 1$ is spun about the line $x = 1$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 2$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 4$ and the $y$ -axis in the first quadrant is spun about the line $x = 4$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 3$ is spun about the line $x = 3$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 1$ . Find the volume.
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = 2$ . Find the volume.

Find volumes by revolving a region
Find the volume when the region bounded by $x = y^{2}$ and $x = 4$ is revolved about the line $x = 4$ .
Find the volume when the region bounded by $y = 2 x$ and $y = 2 x^{2}$ is revolved about the $x$ -axis.
Find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 6$ is revolved about the $x$ -axis.
Find the volume when the region bounded by $x = y^{2}$ and $x = 16$ is revolved about the $y$ -axis.
Find the volume when the region bounded by $y = \sqrt{x}$ and $y = x$ is revolved about the line $y = - 3$ .
Find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 6$ is revolved about the $x$ -axis.



Sigma Prep · sigmaprep.io/worksheets Name Volume by Disks and WashersAnswer keyVersion 1 Date Period Find the volume of each solid. Give exact values. Find volumes by disks and washers Find the volume of the solid formed when the region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 3$ is revolved about the $x$ -axis. $\frac{3376 π}{5}$ Find the volume when the region between $y = x$ and the $y$ -axis, for $0 \leq y \leq 5$ , is revolved about the $y$ -axis. $\frac{125 π}{3}$ The region bounded by $y = 3 x^{2}$ , the $y$ -axis and $y = 8$ is spun about the $y$ -axis. Find the volume. $\frac{32 π}{3}$ The region under $y = \sqrt{4 x}$ from $x = 1$ to $x = 4$ is spun about the $x$ -axis. Find the volume. $30 π$ Find the volume when the region between $y = x^{2}$ and the $y$ -axis, for $0 \leq y \leq 2$ , is revolved about the $y$ -axis. $2 π$ The region bounded by $y = 4 x$ , the $y$ -axis and $y = 9$ is spun about the $y$ -axis. Find the volume. $\frac{243 π}{16}$ Use disks to find the volume of the solid made by rotating the region under $y = 4 x^{2}$ from $x = 1$ to $x = 2$ about the $x$ -axis. $\frac{496 π}{5}$ Find volumes of revolution The region bounded by $y = x^{2}$ , $y = 1$ and the $y$ -axis in the first quadrant is spun about the line $x = 3$ . Find the volume. $\frac{7 π}{2}$ The region bounded by $y = x^{2}$ , $y = 0$ and $x = 1$ is spun about the line $x = 1$ . Find the volume. $\frac{π}{6}$ The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 2$ . Find the volume. $\frac{56 π}{3}$ The region bounded by $y = x^{2}$ , $y = 4$ and the $y$ -axis in the first quadrant is spun about the line $x = 4$ . Find the volume. $\frac{104 π}{3}$ The region bounded by $y = x^{2}$ , $y = 0$ and $x = 3$ is spun about the line $x = 3$ . Find the volume. $\frac{27 π}{2}$ The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 1$ . Find the volume. $\frac{40 π}{3}$ The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = 2$ . Find the volume. $\frac{8 π}{3}$ Find volumes by revolving a region Find the volume when the region bounded by $x = y^{2}$ and $x = 4$ is revolved about the line $x = 4$ . $\frac{512 π}{15}$ Find the volume when the region bounded by $y = 2 x$ and $y = 2 x^{2}$ is revolved about the $x$ -axis. $\frac{8 π}{15}$ Find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 6$ is revolved about the $x$ -axis. $18 π$ Find the volume when the region bounded by $x = y^{2}$ and $x = 16$ is revolved about the $y$ -axis. $\frac{8192 π}{5}$ Find the volume when the region bounded by $y = \sqrt{x}$ and $y = x$ is revolved about the line $y = - 3$ . $\frac{7 π}{6}$ Find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 6$ is revolved about the $x$ -axis. $\frac{5 π}{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Volume by Disks and WashersAnswer keyVersion 1

Date Period

Find the volume of each solid. Give exact values.

Find volumes by disks and washers
Find the volume of the solid formed when the region bounded by $y = 4 x^{2}$ , the $x$ -axis, $x = 2$ and $x = 3$ is revolved about the $x$ -axis. $\frac{3376 π}{5}$
Find the volume when the region between $y = x$ and the $y$ -axis, for $0 \leq y \leq 5$ , is revolved about the $y$ -axis. $\frac{125 π}{3}$
The region bounded by $y = 3 x^{2}$ , the $y$ -axis and $y = 8$ is spun about the $y$ -axis. Find the volume. $\frac{32 π}{3}$
The region under $y = \sqrt{4 x}$ from $x = 1$ to $x = 4$ is spun about the $x$ -axis. Find the volume. $30 π$
Find the volume when the region between $y = x^{2}$ and the $y$ -axis, for $0 \leq y \leq 2$ , is revolved about the $y$ -axis. $2 π$
The region bounded by $y = 4 x$ , the $y$ -axis and $y = 9$ is spun about the $y$ -axis. Find the volume. $\frac{243 π}{16}$
Use disks to find the volume of the solid made by rotating the region under $y = 4 x^{2}$ from $x = 1$ to $x = 2$ about the $x$ -axis. $\frac{496 π}{5}$

Find volumes of revolution
The region bounded by $y = x^{2}$ , $y = 1$ and the $y$ -axis in the first quadrant is spun about the line $x = 3$ . Find the volume. $\frac{7 π}{2}$
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 1$ is spun about the line $x = 1$ . Find the volume. $\frac{π}{6}$
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 2$ . Find the volume. $\frac{56 π}{3}$
The region bounded by $y = x^{2}$ , $y = 4$ and the $y$ -axis in the first quadrant is spun about the line $x = 4$ . Find the volume. $\frac{104 π}{3}$
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 3$ is spun about the line $x = 3$ . Find the volume. $\frac{27 π}{2}$
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = - 1$ . Find the volume. $\frac{40 π}{3}$
The region bounded by $y = x^{2}$ , $y = 0$ and $x = 2$ is spun about the line $x = 2$ . Find the volume. $\frac{8 π}{3}$

Find volumes by revolving a region
Find the volume when the region bounded by $x = y^{2}$ and $x = 4$ is revolved about the line $x = 4$ . $\frac{512 π}{15}$
Find the volume when the region bounded by $y = 2 x$ and $y = 2 x^{2}$ is revolved about the $x$ -axis. $\frac{8 π}{15}$
Find the volume when the region bounded by $y = \sqrt{x}$ , $y = 0$ , and $x = 6$ is revolved about the $x$ -axis. $18 π$
Find the volume when the region bounded by $x = y^{2}$ and $x = 16$ is revolved about the $y$ -axis. $\frac{8192 π}{5}$
Find the volume when the region bounded by $y = \sqrt{x}$ and $y = x$ is revolved about the line $y = - 3$ . $\frac{7 π}{6}$
Find the volume when the region bounded by $y = \frac{1}{x}$ , $y = 0$ , $x = 1$ , and $x = 6$ is revolved about the $x$ -axis. $\frac{5 π}{6}$

How to do these

The region under $y = x$ from $x = 0$ to $x = 3$ is spun about the $x$ -axis. Find the volume.

Each slice is a circle of radius y, which is x.
So the volume is pi times the integral of x squared from 0 to 3.
That is pi times 27 over 3, which is 9 pi.

The answer is $9 π$ .

Where students go wrong

Integrating the radius instead of its square. Each slice is a circle, and the area of a circle is pi r squared, so the function gets squared before it is integrated.

Also called volumes of revolution, disk method, washer method, solid of revolution or rotating about the x-axis.

What you can put on this worksheet

Volume by disks and washers: Find the volume of the solid formed when the region bounded by $y = \sqrt{3 x}$ , the $x$ -axis, $x = 2$ and $x = 5$ is revolved about the $x$ -axis. → $\frac{63 π}{2}$
About other lines, and other curves: The region bounded by $y = \sqrt{x}$ , $y = 0$ and $x = 4$ is spun about the line $y = - 3$ . Find the volume. → $40 π$
Domes, lines and parabolas, other axes, a sphere with a hole: Find the volume when the region bounded by $y = 2 \sqrt{x}$ , $y = 0$ , and $x = 5$ is revolved about the $x$ -axis. → $50 π$

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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.

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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 disks and washers 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 disks and washers, about other lines, and other curves and domes, lines and parabolas, other axes, a sphere with a hole. Tick as many as you want and set how many of each, or let it spread them evenly.

Integrating the radius instead of its square. Each slice is a circle, and the area of a circle is pi r squared, so the function gets squared before it is integrated.

Volume by disks and washers worksheet

Select your difficulty

Find volumes by disks and washers

Find volumes of revolution

Find volumes by revolving a region

Find volumes by disks and washers

Find volumes of revolution

Find volumes by revolving a region

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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