17Calculus Integrals - Surface Area
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.]
Recommended Books on Amazon | ||
|---|---|---|
  |
  |
  |
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 |
|---|
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.
1195 video
video by PatrickJMT |
|---|
1195 video
video by PatrickJMT |
|---|
|
close solution
|
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 |
|---|
1196 video
video by Krista King Math |
|---|
|
close solution
|
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 |
|---|
1197 video
video by Krista King Math |
|---|
|
close solution
|
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 |
|---|
1198 video
video by MIP4U |
|---|
|
close solution
|
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 |
|---|
1199 video
video by MIP4U |
|---|
|
close solution
|
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 |
|---|
1915 video
video by MIP4U |
|---|
|
close solution
|
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 |
|---|
2002 video
video by Dr Chris Tisdell |
|---|
Final Answer |
|---|
\([(17)^{3/2}-1]\pi/6\) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
You CAN Ace Calculus
Topics You Need To Understand For This Page
Related Topics and Links
external links you may find helpful |
|---|
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
|
Set 1 - basic identities | |||
|---|---|---|---|
\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
|
Set 2 - squared identities | ||
|---|---|---|
\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
|
Set 3 - double-angle formulas | |
|---|---|
\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
|
Set 4 - half-angle formulas | |
|---|---|
\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
To bookmark this page and practice problems, log in to your account or set up a free account.
Topics Listed Alphabetically
Single Variable Calculus |
|---|
Multi-Variable Calculus |
|---|
Differential Equations |
|---|
Precalculus |
|---|
Engineering |
|---|
Circuits |
|---|
Semiconductors |
|---|
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.
|
| |
The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. |
|
|---|
Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com! |
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.
