17Calculus Polar Coordinates - Area
17Calculus
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.
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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 |
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Practice
Unless otherwise instructed, calculate these areas, giving your answers in exact, simplified form.
Basic |
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\(r=2\cos\theta\)
Problem Statement
Calculate the area enclosed by the polar curve \(r=2\cos\theta\).
Solution
1383 video solution
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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 solution
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one loop of \(r=\sin(4\theta)\)
Problem Statement
Calculate the area enclosed by one loop of \(r=\sin(4\theta)\).
Solution
1272 video solution
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\(r=3+\sin\theta\)
Problem Statement
Calculate the area enclosed by \(r=3+\sin\theta\).
Solution
1276 video solution
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Intermediate |
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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 solution
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inside both \(r=\cos\theta\) and \(r=\sin(2\theta)\)
Problem Statement |
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Calculate the area inside both \(r=\cos\theta\) and \(r=\sin(2\theta)\).
Final Answer |
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\(\displaystyle{ \frac{\pi}{4} - \frac{3\sqrt{3}}{16} }\)
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 solution
Final Answer
\(\displaystyle{ \frac{\pi}{4} - \frac{3\sqrt{3}}{16} }\)
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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 solution
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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 solution
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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 solution
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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 solution
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