Consider the region $R$ in the first quadrant that is outside the circle $r = 1$ and inside the four-leaved rose $r = 2sin2\theta$.

Write the following double integral as an iterated integral in polar coordinates: $$\iint cos2\theta\, dA$$

To find the interval of $\theta$, I set $2sin2\theta = 1$, getting $\frac{\pi}{12}$ and $\frac{5\pi}{12}$. To find the interval of $r$, I got 1 and $2sin2\theta$ based off of the description of region $R$. Would this mean that my double iterated integral is simply: $$\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \int_{1}^{2sin2\theta} cos2\theta\, drd\theta$$

Do I need to do anything in regards to converting the initial double integral to polar form? Or is this fine?

  • 2
    $\begingroup$ Yes, that's correct. $\endgroup$ – Andrew Li Jul 2 '18 at 0:14

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