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

17Calculus Parametric Equations - Surface Area

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Calculate the area of a surface that is described by parametric equations.

Equations

To calculate the surface area defined by revolving a parametric curve defined as \( x=X(t)\) and \(y=Y(t)\) from \(t=t_0\) to \(t=t_1\) about the x-axis, we use this integral. \[ S = 2\pi ~ \int_{t_0}^{t_1}{ Y(t) \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~ dt} \] If you compare this integral to the equation for arc length (on the prevous page) you will see the common factor \( \sqrt{[X'(t)]^2 + [Y'(t)]^2} \)

Some textbooks write the surface integral differently, taking this into account. You may see it written as \[ S = 2\pi ~ \int_{t_0}^{t_1}{Y(t) ~ ds} \] where \( ds = \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~dt \)

When rotating about the y-axis, the integral we use is \[ S = 2\pi ~ \int_{t_0}^{t_1}{X(t) ~ ds} \] Notice we use a capital \(S\) to represent surface area. This is the standard symbol you will see in many textbooks. We reserve a lowercase \(s\) to represent arc length.

See the resources section for links to pages explaining the derivation of these equations.

Schaum's Outline of Calculus, 6e: 1,105 Solved Problems + 30 Videos

Practice

Unless otherwise instructed, calculate the surface area formed by revolving the parametric curve about the given axis.

\(x = t^4+4\), \(y = 8t\), \( 0 \leq t \leq 2 \); \(x\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \(x = t^4+4\), \(y = 8t\), \( 0 \leq t \leq 2 \) rotated about the \(x\)-axis.

Solution

In this video, when he sets up his integral, he initially places the \(Y(t)\) term outside the integral. Then he moves the \(t\) inside the integral to integrate. These steps are not correct. The \(Y(t)\) term should be inside the integral the entire time. He does eventually stumble into the correct answer but his procedure is not correct.

The Organic Chemistry Tutor - 3507 video solution

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

\(x = t^2\), \(y = t^3\), \(0 \leq t \leq 1\); \(y\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \(x = t^2\), \(y = t^3\), \(0 \leq t \leq 1\) rotated about the \(y\)-axis.

Solution

In this video, when he sets up his integral, he initially places the \(X(t)\) term outside the integral. Then he moves the \(t\) inside the integral to integrate. These steps are not correct. The \(X(t)\) term should be inside the integral the entire time. He does eventually stumble into the correct answer but his procedure is not correct.

The Organic Chemistry Tutor - 3508 video solution

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

\(x = 3t^2\), \(y = 2t^3\), \(0 \leq t \leq 5\); \(y\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \(x = 3t^2\), \(y = 2t^3\), \(0 \leq t \leq 5\) rotated about the \(y\)-axis.

Solution

Krista King Math - 594 video solution

video by Krista King Math

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

\(x = 1-t\), \(y = 2\sqrt{t}\), \(1 \leq t \leq 4\); \(x\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \(x = 1-t\), \(y = 2\sqrt{t}\), \(1 \leq t \leq 4\) rotated about the \(x\)-axis.

Solution

Krista King Math - 59 video solution

video by Krista King Math

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

\( x = 3 + 2t \), \( y = 9 - 3t \), \( 1 \leq t \leq 4 \); \(y\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \( x = 3 + 2t \), \( y = 9 - 3t \), \( 1 \leq t \leq 4 \) rotated about the \(y\)-axis.

Solution

The solution can be found on this page.

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

\( x = 9 + 2t^2 \), \( y = 4t \), \( 0 \leq t \leq 2 \); \(x\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \( x = 9 + 2t^2 \), \( y = 4t \), \( 0 \leq t \leq 2 \) rotated about the \(x\)-axis.

Solution

The solution can be found on this page.

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

\( x = 3\cos(\pi t) \), \( y = 5t + 2 \), \( 0 \leq t \leq 1/2 \); \(y\)-axis

Problem Statement

Find the surface area of revolution of the parametric curve \( x = 3\cos(\pi t) \), \( y = 5t + 2 \), \( 0 \leq t \leq 1/2 \) rotated about the \(y\)-axis.

Solution

The solution can be found on this page.

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

Set up, but do not evaluate, an integral that calculates the surface area of revolution of the curve \( x = 1 + \ln(5 + t^2) \), \( y = 2t - 2t^2 \), \( 0 \leq t \leq 2 \) rotated about the \(x\)-axis.

Problem Statement

Set up, but do not evaluate, an integral that calculates the surface area of revolution of the curve \( x = 1 + \ln(5 + t^2) \), \( y = 2t - 2t^2 \), \( 0 \leq t \leq 2 \) rotated about the \(x\)-axis.

Solution

The solution can be found on this page.

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

Set up, but do not evaluate, an integral that calculates the surface area of revolution of the curve \( x = 1 + 3t^2 \), \( y = \sin(2t)\cos(t/4) \), \( 0 \leq t \leq 1/2 \) rotated about the \(y\)-axis.

Problem Statement

Set up, but do not evaluate, an integral that calculates the surface area of revolution of the curve \( x = 1 + 3t^2 \), \( y = \sin(2t)\cos(t/4) \), \( 0 \leq t \leq 1/2 \) rotated about the \(y\)-axis.

Solution

The solution can be found on this page.

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

Really UNDERSTAND Calculus

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

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.

math and science learning techniques

Join Amazon Prime - Watch Thousands of Movies & TV Shows Anytime - Start Free Trial Now

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Practice Search

Practice Instructions

Unless otherwise instructed, calculate the surface area formed by revolving the parametric curve about the given axis.

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