Convert to polar $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)}\, \mathrm dy \, \mathrm dx$

Question

I have the double integral

$\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)}\, \mathrm dy \, \mathrm dx$

And I am asked to evaluate this by changing to polar coordinates.

I know the transformations are, $x=r \cos\phi$, $y=r \sin \phi$ and $dx ~dy=r ~dr~ d\phi$

I am just confused about:

• As I have $dy~dx$ in the question, do I need to change $r~dr~d\phi$?
• Do I have to change the limits of integration, if so how?

Answer 1

Geometrically, we have that $0 \le x \le a$ and $0 \le y \le \sqrt{a^2 - x^2}$; rearranging this leads to

$$0 \le y^2 + x^2 \le a^2$$

So this is a quarter circle of radius $a$, in the first quadrant. So the appropriate bounds will be $0 \le r \le a$ and $0 \le \theta \le \frac{\pi}{2}$.

To make the change into polar coordinates, the differential area is now $r dr d\theta$ instead of $dx dy$, so we simply erase one and write the other. Hence you should find

$$\int_0^{\pi/2} \int_0^{a} e^{-r^2 - a^2} r dr d\theta $$