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I know that the area of a curve given in polar coordinates is $$\int_{\theta_1}^{\theta_2}\frac{r^2}{2}d\theta$$. But what is the area outside one curve and inside another, when one of them is not entirely inside the other? For example, what is the area inside $a(1+\cos{\theta})$ and outside $a\sin{\theta}$?

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    $\begingroup$ find the coordinates of intersection and use those as limits of the integration, integrate the two and subtractone from the other $\endgroup$ – danimal Jun 16 '15 at 8:17
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    $\begingroup$ it is essential to draw a picture and determine the limits of integration $\endgroup$ – David Quinn Jun 16 '15 at 8:29
  • $\begingroup$ So the two curves intersect on pi/2 for the first time. And I calculate the area of the cardioid up to pi/2, then I subtract the area of half the circle. I think I got it. Thank you both. $\endgroup$ – Laxuist Jun 16 '15 at 9:28
  • $\begingroup$ would you mind explaining why the above formula makes sense when it comes to calculating the area of a enclosed curve in polar coordinates?? I am a little confused about the formula.... $\endgroup$ – NiubilityDiu Jul 20 '15 at 20:49
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integrate the cardioid from $-\pi$ to $\frac {\pi}{2}$ and subtract half the circle

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  • $\begingroup$ Thank you, I did so. May I ask how did you make the plot in the image? $\endgroup$ – Laxuist Jun 16 '15 at 10:12
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    $\begingroup$ I plotted the cardiod using uk.mathworks.com/help/symbolic/ezpolar.html and then imported it into word, inserted a circle and saved it as a png $\endgroup$ – David Quinn Jun 16 '15 at 10:18
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    $\begingroup$ sounds like a lot of work! try desmos instead :) desmos.com/calculator/va717uzuwb $\endgroup$ – danimal Jun 16 '15 at 10:50
  • $\begingroup$ @danimal: it only took about a minute, but thanks for this link - it looks really good! $\endgroup$ – David Quinn Jun 16 '15 at 10:53

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