Question about Integrating in Polar Coordinates

Question

If you have $$ \int_0^2 \int_0^{\sqrt{4 - x^2}} e^{-(x^2 + y^2)} dy \, dx $$ and you convert to polar coordinates, you integrate from $0$ to $\pi/2$) with respect to theta.

But, if you have $$ \int_{-6}^6 \int_0^{\sqrt{36-x^2}} \sin(x^2+y^2) \, dy \, dx $$ and you convert to polar coordinates, you integrate from $0$ to $\pi$ with respect to theta. Can someone explain to me why the bounds of integration with respect to theta are different in these two problems? I'm having a hard time figuring it out. It would be a lot of help. Thanks.

Answer 1

Sometimes a picture is worth 1000 words:

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Graphics[
 {Red, Disk[{0, 0}, 2, {0, \[Pi]/2}],
  Opacity[0.1], Green, Disk[{0, 0}, 6, {0, \[Pi]}]},
 Axes -> Automatic]

Answer 2

Let's construct formal proof. In first case we have set $$\left\lbrace \begin{array}{l}0 \leqslant x \leqslant 2 \\ 0 \leqslant y \leqslant \sqrt{4-x^2} \end{array}\right\rbrace$$ considering polar coordinates $x=r\cos \theta, y=r\sin \theta$ we will have from first inequalities $0 \leqslant r\cos \theta \leqslant 2 $ and $0 \leqslant r\sin \theta \leqslant \sqrt{4-(r\cos \theta)^2}$. From here we have for $\theta $ inequalities $0 \leqslant \sin \theta$ and $0 \leqslant \cos \theta$ which gives $\theta \in \left[0, \frac{\pi}{2}\right]$. After analysis inequalities $r \leqslant \frac{2}{\cos \theta}$ and $r \leqslant 2$ we have set

$$\left\lbrace \begin{array}{l}0 \leqslant \theta \leqslant \frac{\pi}{2} \\ 0 \leqslant r \leqslant 2 \end{array}\right\rbrace$$ For second case analogical analysis gives

$$\left\lbrace \begin{array}{l}-6 \leqslant x \leqslant 6 \\ 0 \leqslant y \leqslant \sqrt{36-x^2} \end{array}\right\rbrace \to \left\lbrace \begin{array}{l}0 \leqslant \theta \leqslant \pi \\ 0 \leqslant r \leqslant 6 \end{array}\right\rbrace$$ As clevely is written in adjacent answer from David G. Stork "Sometimes a picture is worth 1000 words". But the most best picture is not mathematical proof - good picture helps us to construct correct mathematical proof.