[Math] How to define the limits of a double integral in polar coordinates over an annulus

multivariable-calculuspolar coordinates

Evaluate the double integral by re-writing them in polar coordinates:

$\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying in the first quadrant and below the line $y=x$.

So from this, I gather (assuming I understood correctly) that $R=\{(x,y)\mid9\leq x^2+y^2\leq 25,\ 0\leq y\leq x\}$.

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There are two circles, one where $r=5$ and one where $r=3$. I would guess that what I'm looking for is the larger circle minus the smaller circle, but doing that the only way I can think of yields the following:

\begin{gather}
(x^2+y^2)-(x^2+y^2)=25-9\\
0=16
\end{gather}

I'm clearly way off track here; how do I look at this so I can define the bounds of the double integral in polar coordinates?

Best Answer

You can use polar coordinates, then you have

$$ 9\leq x^2+y^2\leq 25 \implies 3\leq r\leq 5, $$

and

$$ 0\leq \theta \leq \frac{\pi}{4},\quad \rm{since}\quad y=x. $$

$$ \displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA = \int_{0}^{\pi/4}\int_{3}^{5}\frac{\sin^{2}(\theta)}{\cos^2(\theta)}rdrd\theta $$

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