Polar Integration

**flybynight** · October 4th 2010, 04:28 PM

I am given two functions: $r=1-\sin\theta$ and $r=0.5$ . I am asked to find the area inside the cardioid and outside of the circle. How should I proceed? Are there any good resources for these types of problems?

Thanks,
Peter

**skeeter** · October 4th 2010, 06:54 PM

Originally Posted by flybynight

I am given two functions: $r=1-\sin\theta$ and $r=0.5$ . I am asked to find the area inside the cardioid and outside of the circle. How should I proceed? Are there any good resources for these types of problems?

Thanks,
Peter

note the cardiod and circle intersect at $\displaystyle \theta = \frac{\pi}{6}$ in quad I.

using the symmetry of the y-axis ...

$\displaystyle A = 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{6}} \frac{(1-\sin{\theta})^2}{2} - \frac{1}{8} \, d\theta$

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