Surface area of revolution worksheet

Each strip of the curve sweeps out a thin band whose area is its circumference times its slant length, 2πr ds. Adding the bands is an integral, and the curves here are chosen so the square root in ds simplifies and the answer comes out exact.

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Sigma Prep · sigmaprep.io/worksheets Name Surface Area of Revolution Date Period Find the area of each surface. Give exact answers. Find the area of the surface made by rotating $y = \sqrt{4 - x^{2}}$ , $0 \leq x \leq 2$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{x^{2}}{2}$ , $0 \leq x \leq \sqrt{15}$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \frac{3 x}{4} + 2$ , $0 \leq x \leq 4$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{3 x}{4}$ , $3 \leq x \leq 5$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \sqrt{2 x}$ , $4 \leq x \leq 24$ , about the $x$ -axis. Find the area of the surface made by rotating $y = x^{2}$ , $0 \leq x \leq \sqrt{6}$ , about the $y$ -axis. Find the area of the surface made by rotating $y = 2 \sqrt{2 x}$ , $0 \leq x \leq 16$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{4 x}{3} + 2$ , $3 \leq x \leq 7$ , about the $y$ -axis. Find the area of the surface made by rotating $y = 2 \sqrt{x}$ , $24 \leq x \leq 35$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{3 x}{4} - 2$ , $1 \leq x \leq 3$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \sqrt{49 - x^{2}}$ , $- 6 \leq x \leq 1$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{4 x}{3}$ , $0 \leq x \leq 5$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \sqrt{64 - x^{2}}$ , $- 7 \leq x \leq - 6$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{8 x}{15} - 3$ , $1 \leq x \leq 2$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \frac{12 x}{5} + 2$ , $5 \leq x \leq 15$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{15 x}{8} + 1$ , $3 \leq x \leq 5$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \sqrt{81 - x^{2}}$ , $- 5 \leq x \leq 9$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{5 x}{12}$ , $0 \leq x \leq 3$ , about the $y$ -axis. Find the area of the surface made by rotating $y = \sqrt{6 x}$ , $12 \leq x \leq 36$ , about the $x$ -axis. Find the area of the surface made by rotating $y = \frac{5 x}{12} + 1$ , $1 \leq x \leq 3$ , about the $y$ -axis. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Surface Area of Revolution

Date Period

Find the area of each surface. Give exact answers.

Find the area of the surface made by rotating $y = \sqrt{4 - x^{2}}$ , $0 \leq x \leq 2$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{x^{2}}{2}$ , $0 \leq x \leq \sqrt{15}$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \frac{3 x}{4} + 2$ , $0 \leq x \leq 4$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{3 x}{4}$ , $3 \leq x \leq 5$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \sqrt{2 x}$ , $4 \leq x \leq 24$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = x^{2}$ , $0 \leq x \leq \sqrt{6}$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = 2 \sqrt{2 x}$ , $0 \leq x \leq 16$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{4 x}{3} + 2$ , $3 \leq x \leq 7$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = 2 \sqrt{x}$ , $24 \leq x \leq 35$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{3 x}{4} - 2$ , $1 \leq x \leq 3$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \sqrt{49 - x^{2}}$ , $- 6 \leq x \leq 1$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{4 x}{3}$ , $0 \leq x \leq 5$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \sqrt{64 - x^{2}}$ , $- 7 \leq x \leq - 6$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{8 x}{15} - 3$ , $1 \leq x \leq 2$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \frac{12 x}{5} + 2$ , $5 \leq x \leq 15$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{15 x}{8} + 1$ , $3 \leq x \leq 5$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \sqrt{81 - x^{2}}$ , $- 5 \leq x \leq 9$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{5 x}{12}$ , $0 \leq x \leq 3$ , about the $y$ -axis.
Find the area of the surface made by rotating $y = \sqrt{6 x}$ , $12 \leq x \leq 36$ , about the $x$ -axis.
Find the area of the surface made by rotating $y = \frac{5 x}{12} + 1$ , $1 \leq x \leq 3$ , about the $y$ -axis.



Sigma Prep · sigmaprep.io/worksheets Name Surface Area of RevolutionAnswer keyVersion 1 Date Period Find the area of each surface. Give exact answers. Find the area of the surface made by rotating $y = \sqrt{4 - x^{2}}$ , $0 \leq x \leq 2$ , about the $x$ -axis. $8 π$ Find the area of the surface made by rotating $y = \frac{x^{2}}{2}$ , $0 \leq x \leq \sqrt{15}$ , about the $y$ -axis. $42 π$ Find the area of the surface made by rotating $y = \frac{3 x}{4} + 2$ , $0 \leq x \leq 4$ , about the $x$ -axis. $35 π$ Find the area of the surface made by rotating $y = \frac{3 x}{4}$ , $3 \leq x \leq 5$ , about the $y$ -axis. $20 π$ Find the area of the surface made by rotating $y = \sqrt{2 x}$ , $4 \leq x \leq 24$ , about the $x$ -axis. $\frac{632 π}{3}$ Find the area of the surface made by rotating $y = x^{2}$ , $0 \leq x \leq \sqrt{6}$ , about the $y$ -axis. $\frac{62 π}{3}$ Find the area of the surface made by rotating $y = 2 \sqrt{2 x}$ , $0 \leq x \leq 16$ , about the $x$ -axis. $\frac{832 π}{3}$ Find the area of the surface made by rotating $y = \frac{4 x}{3} + 2$ , $3 \leq x \leq 7$ , about the $y$ -axis. $\frac{200 π}{3}$ Find the area of the surface made by rotating $y = 2 \sqrt{x}$ , $24 \leq x \leq 35$ , about the $x$ -axis. $\frac{728 π}{3}$ Find the area of the surface made by rotating $y = \frac{3 x}{4} - 2$ , $1 \leq x \leq 3$ , about the $y$ -axis. $10 π$ Find the area of the surface made by rotating $y = \sqrt{49 - x^{2}}$ , $- 6 \leq x \leq 1$ , about the $x$ -axis. $98 π$ Find the area of the surface made by rotating $y = \frac{4 x}{3}$ , $0 \leq x \leq 5$ , about the $y$ -axis. $\frac{125 π}{3}$ Find the area of the surface made by rotating $y = \sqrt{64 - x^{2}}$ , $- 7 \leq x \leq - 6$ , about the $x$ -axis. $16 π$ Find the area of the surface made by rotating $y = \frac{8 x}{15} - 3$ , $1 \leq x \leq 2$ , about the $y$ -axis. $\frac{17 π}{5}$ Find the area of the surface made by rotating $y = \frac{12 x}{5} + 2$ , $5 \leq x \leq 15$ , about the $x$ -axis. $1352 π$ Find the area of the surface made by rotating $y = \frac{15 x}{8} + 1$ , $3 \leq x \leq 5$ , about the $y$ -axis. $34 π$ Find the area of the surface made by rotating $y = \sqrt{81 - x^{2}}$ , $- 5 \leq x \leq 9$ , about the $x$ -axis. $252 π$ Find the area of the surface made by rotating $y = \frac{5 x}{12}$ , $0 \leq x \leq 3$ , about the $y$ -axis. $\frac{39 π}{4}$ Find the area of the surface made by rotating $y = \sqrt{6 x}$ , $12 \leq x \leq 36$ , about the $x$ -axis. $588 π$ Find the area of the surface made by rotating $y = \frac{5 x}{12} + 1$ , $1 \leq x \leq 3$ , about the $y$ -axis. $\frac{26 π}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Surface Area of RevolutionAnswer keyVersion 1

Date Period

Find the area of each surface. Give exact answers.

Find the area of the surface made by rotating $y = \sqrt{4 - x^{2}}$ , $0 \leq x \leq 2$ , about the $x$ -axis. $8 π$
Find the area of the surface made by rotating $y = \frac{x^{2}}{2}$ , $0 \leq x \leq \sqrt{15}$ , about the $y$ -axis. $42 π$
Find the area of the surface made by rotating $y = \frac{3 x}{4} + 2$ , $0 \leq x \leq 4$ , about the $x$ -axis. $35 π$
Find the area of the surface made by rotating $y = \frac{3 x}{4}$ , $3 \leq x \leq 5$ , about the $y$ -axis. $20 π$
Find the area of the surface made by rotating $y = \sqrt{2 x}$ , $4 \leq x \leq 24$ , about the $x$ -axis. $\frac{632 π}{3}$
Find the area of the surface made by rotating $y = x^{2}$ , $0 \leq x \leq \sqrt{6}$ , about the $y$ -axis. $\frac{62 π}{3}$
Find the area of the surface made by rotating $y = 2 \sqrt{2 x}$ , $0 \leq x \leq 16$ , about the $x$ -axis. $\frac{832 π}{3}$
Find the area of the surface made by rotating $y = \frac{4 x}{3} + 2$ , $3 \leq x \leq 7$ , about the $y$ -axis. $\frac{200 π}{3}$
Find the area of the surface made by rotating $y = 2 \sqrt{x}$ , $24 \leq x \leq 35$ , about the $x$ -axis. $\frac{728 π}{3}$
Find the area of the surface made by rotating $y = \frac{3 x}{4} - 2$ , $1 \leq x \leq 3$ , about the $y$ -axis. $10 π$
Find the area of the surface made by rotating $y = \sqrt{49 - x^{2}}$ , $- 6 \leq x \leq 1$ , about the $x$ -axis. $98 π$
Find the area of the surface made by rotating $y = \frac{4 x}{3}$ , $0 \leq x \leq 5$ , about the $y$ -axis. $\frac{125 π}{3}$
Find the area of the surface made by rotating $y = \sqrt{64 - x^{2}}$ , $- 7 \leq x \leq - 6$ , about the $x$ -axis. $16 π$
Find the area of the surface made by rotating $y = \frac{8 x}{15} - 3$ , $1 \leq x \leq 2$ , about the $y$ -axis. $\frac{17 π}{5}$
Find the area of the surface made by rotating $y = \frac{12 x}{5} + 2$ , $5 \leq x \leq 15$ , about the $x$ -axis. $1352 π$
Find the area of the surface made by rotating $y = \frac{15 x}{8} + 1$ , $3 \leq x \leq 5$ , about the $y$ -axis. $34 π$
Find the area of the surface made by rotating $y = \sqrt{81 - x^{2}}$ , $- 5 \leq x \leq 9$ , about the $x$ -axis. $252 π$
Find the area of the surface made by rotating $y = \frac{5 x}{12}$ , $0 \leq x \leq 3$ , about the $y$ -axis. $\frac{39 π}{4}$
Find the area of the surface made by rotating $y = \sqrt{6 x}$ , $12 \leq x \leq 36$ , about the $x$ -axis. $588 π$
Find the area of the surface made by rotating $y = \frac{5 x}{12} + 1$ , $1 \leq x \leq 3$ , about the $y$ -axis. $\frac{26 π}{3}$

How to do these

Find the area of the surface made by rotating $y = 2 \sqrt{x}$ , $0 \leq x \leq 3$ , about the $x$ -axis.

y' = 1/√x, so 1 + (y')² = (x + 1)/x and ds = √((x + 1)/x) dx.
The radius is y = 2√x, so 2πy ds = 2π · 2√x · √((x + 1)/x) dx = 4π√(x + 1) dx.
S = 4π ∫ from 0 to 3 of √(x + 1) dx = 4π · (2/3)[(x + 1)^(3/2)] from 0 to 3 = (8π/3)(8 - 1).

The answer is $\frac{56 π}{3}$ .

Where students go wrong

Using dx in place of ds. That is the volume formula’s habit: the surface needs the slant length √(1 + (y′)²) dx, and leaving it out gives too small an answer for any curve that is not flat.

Also called area of a surface of revolution, surface area by integration or surface area of a solid of revolution.

What you can put on this worksheet

About the x-axis: Find the area of the surface made by rotating $y = \frac{8 x}{15}$ , $15 \leq x \leq 45$ , about the $x$ -axis. → $1088 π$
About the y-axis: Find the area of the surface made by rotating $y = \frac{12 x}{5} + 2$ , $1 \leq x \leq 3$ , about the $y$ -axis. → $\frac{104 π}{5}$

Questions about these worksheets

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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 surface area of revolution 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 2: about the x-axis and about the y-axis. Tick as many as you want and set how many of each, or let it spread them evenly.

Using dx in place of ds. That is the volume formula’s habit: the surface needs the slant length √(1 + (y′)²) dx, and leaving it out gives too small an answer for any curve that is not flat.

Surface area of revolution worksheet

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Surface Area of Revolution

Surface Area of RevolutionAnswer keyVersion 1

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