\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Integrals - Surface Area

17Calculus

Limits

Using Limits


Derivatives

Graphing

Related Rates

Optimization

Other Applications

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

Infinite Series

Applications

Tools

Parametrics

Conics

Polar Coordinates

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

Learning Tools

Articles

This page covers the topic of surface area of an explicitly defined smooth curve revolved around an axis in the xy-plane in cartesian (rectangular) coordinates. [You can also calculate surface area in polar coordinates and for surfaces described parametrically.]

Setting up the integral to calculate surface area is fairly straight-forward. The difficulty with this topic occurs when evaluating the integral, which can quickly become quite complicated. Consequently, most problems you get will be carefully hand-picked by your instructor or the textbook author so that you can evaluate the integrals with the techniques you know. The comments we made on the arc length page about tricks to evaluating these integrals apply here as well.

First, let's look at a video clip explaining how to derive the surface area equations.

MIP4U - Surface Area of Revolution - Part 1 of 2 [5min-16secs]

video by MIP4U

rotation about the x-axis

\(\int_{a}^{b}{2\pi y ~ ds}\)

rotation about the y-axis

\(\int_{c}^{d}{2\pi x ~ ds}\)

\(ds = \sqrt{1 + [f'(x)]^2} ~dx \) or \( ds = \sqrt{1 + [g'(y)]^2} ~dy \)

Notes
1. \(ds\) in the last row above is the differential arc length as discussed on the arc length page. Using \(ds\) allows us to write the integral in a more compact form and it is easier to see what is going on.
2. Which \(ds\) you use depends on how the graph is described.

Here is a great video clip explaining these equations in more detail.

PatrickJMT - Finding Surface Area - Part 1 [1min-2secs]

video by PatrickJMT

Math Word Problems Demystified

Practice

Unless otherwise instructed, calculate the surface area of the given line segment rotated about the given axis. Give all answers in exact form.

equation: \(y=\sqrt{x}\)

range: \(4\leq x\leq9\)

axis of rotation: x-axis

Problem Statement

Calculate the surface area of the line segment \(y=\sqrt{x}\); \(4\leq x\leq9\) rotated about the x-axis. Give your answer in exact form.

Solution

He works this problem twice in two videos, using different \(ds\) equations.

PatrickJMT - 1195 video solution

video by PatrickJMT

PatrickJMT - 1195 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

equation: \(y=x^2-\frac{1}{8}\ln(x)\)

range: \(1\leq x\leq2\)

axis of rotation: y-axis

Problem Statement

Calculate the surface area of the line segment \(y=x^2-\frac{1}{8}\ln(x)\); \(1\leq x\leq2\) rotated about the y-axis. Give your answer in exact form.

Solution

Krista King Math - 1196 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

equation: \(y=\sqrt{4-x^2}\)

range: \(-1\leq x\leq1\)

axis of rotation: x-axis

Problem Statement

Calculate the surface area of the line segment \(y=\sqrt{4-x^2}\); \(-1\leq x\leq1\) rotated about the x-axis. Give your answer in exact form.

Solution

Krista King Math - 1197 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

equation: \(f(x)=(1/3)x^3\)

range: \([0,2]\)

axis of rotation: x-axis

Problem Statement

Calculate the surface area of the line segment \(f(x)=(1/3)x^3\); \([0,2]\) rotated about the x-axis. Give your answer in exact form.

Solution

MIP4U - 1198 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

equation: \(f(x)=\sqrt[3]{x}\)

range: \([0,8]\)

axis of rotation: y-axis

Problem Statement

Calculate the surface area of the line segment \(f(x)=\sqrt[3]{x}\); \([0,8]\) rotated about the y-axis. Give your answer in exact form.

Solution

MIP4U - 1199 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

equation: \(y=\sqrt[3]{6x}\)

range: \(0 \leq x \leq 4/3\)

axis of rotation: y-axis

Problem Statement

Calculate the surface area of the line segment \(y=\sqrt[3]{6x}\); \(0 \leq x \leq 4/3\) rotated about the y-axis. Give your answer in exact form.

Solution

MIP4U - 1915 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

equation: \(f(x)=\sqrt{x}\)

range: \(0\leq x\leq 4\)

axis of rotation: x-axis

Problem Statement

Calculate the surface area of the line segment \(f(x)=\sqrt{x}\); \(0\leq x\leq 4\) rotated about the x-axis. Give your answer in exact form.

Final Answer

\([(17)^{3/2}-1]\pi/6\)

Problem Statement

Calculate the surface area of the line segment \(f(x)=\sqrt{x}\); \(0\leq x\leq 4\) rotated about the x-axis. Give your answer in exact form.

Solution

Dr Chris Tisdell - 2002 video solution

video by Dr Chris Tisdell

Final Answer

\([(17)^{3/2}-1]\pi/6\)

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Calculus

Topics You Need To Understand For This Page

Related Topics and Links

external links you may find helpful

surface area youtube playlist

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

effective study techniques

Shop Amazon - New Textbooks - Save up to 40%

As an Amazon Associate I earn from qualifying purchases.

Practice Instructions

Unless otherwise instructed, calculate the surface area of the given line segment rotated about the given axis. Give all answers in exact form.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics