17Calculus Polar Coordinates - Area
On this page we cover a common calculus problem involving polar coordinates, determining area.
As mentioned on the main polar coordinates page, polar coordinates are just parametric equations. If you are familiar with parametric equations, this material should be very intuitive.
Recommended Books on Amazon | ||
|---|---|---|
  |
  |
  |
To calculate the area defined by a polar curve r, which is a function of \(\theta\), between the rays \(\theta = \theta_0\) and \(\theta = \theta_1\), we use the integral \[ A = \frac{1}{2} \int_{\theta_0}^{\theta_1}{ r^2 d\theta} \] Here is a good in-depth video discussing area using polar coordinates.
MIT OCW - Lec 33 | MIT 18.01 Single Variable Calculus, Fall 2007 [35min]
video by MIT OCW |
|---|
Practice
Unless otherwise instructed, calculate these areas, giving your answers in exact, simplified form.
Basic |
|---|
\(r=2\cos\theta\)
Problem Statement |
|---|
Calculate the area enclosed by the polar curve \(r=2\cos\theta\).
Solution |
|---|
1383 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
one loop of \(r=2\cos(4\theta)\)
Problem Statement |
|---|
Calculate the area bounded by one loop of \(r=2\cos(4\theta)\).
Solution |
|---|
1384 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
one loop of \(r=\sin(4\theta)\)
Problem Statement |
|---|
Calculate the area enclosed by one loop of \(r=\sin(4\theta)\).
Solution |
|---|
1272 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
\(r=3+\sin\theta\)
Problem Statement |
|---|
Calculate the area enclosed by \(r=3+\sin\theta\).
Solution |
|---|
1276 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Intermediate |
|---|
inside \(r=1+2\cos(\theta)\) and outside \(r=2\)
Problem Statement |
|---|
Calculate the area inside \(r=1+2\cos(\theta)\) and outside \(r=2\).
Solution |
|---|
1382 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
inside both \(r=\cos\theta\) and \(r=\sin(2\theta)\)
Problem Statement |
|---|
Calculate the area inside both \(r=\cos\theta\) and \(r=\sin(2\theta)\).
Final Answer |
|---|
\(\displaystyle{ \frac{\pi}{4} - \frac{3\sqrt{3}}{16} }\)
Problem Statement |
|---|
Calculate the area inside both \(r=\cos\theta\) and \(r=\sin(2\theta)\).
Solution |
|---|
The final answer at the end of the video is incorrect. At about the 11:26 mark he notes his mistake.
1273 video
Final Answer |
|---|
\(\displaystyle{ \frac{\pi}{4} - \frac{3\sqrt{3}}{16} }\) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
inside \(r=1+\sin\theta\) and outside \(r=1\)
Problem Statement |
|---|
Calculate the area that lies inside \(r=1+\sin\theta\) and outside \(r=1\).
Solution |
|---|
1274 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
inside both \(r=9\cos\theta\) and \(r=9\sin\theta\)
Problem Statement |
|---|
Calculate the area of the region inside both \(r=9\cos\theta\) and \(r=9\sin\theta\).
Solution |
|---|
1275 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
outside \(r=\sin\theta\) and inside \(r=2\sin\theta\)
Problem Statement |
|---|
Calculate the area outside \(r=\sin\theta\) and inside \(r=2\sin\theta\).
Solution |
|---|
1277 video
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
inside both \(r=\sin(2\theta)\) and \(r=\cos(2\theta)\)
Problem Statement |
|---|
Calculate the area of the region inside both \(r=\sin(2\theta)\) and \(r=\cos(2\theta)\).
Solution |
|---|
1278 video
|
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
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 |
|---|
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. |
|
|---|
Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more! |
Practice Instructions
Unless otherwise instructed, calculate these areas, giving your answers in exact, simplified form.
