use polar coordinates to evaluate the following integral
$\int^{a}_{-a} \int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2)}dydx$
i did in my paper was
my limit are after converting it to polar coordinates
$\int^{\pi}_0 \int^{{a}}_0 e^{-r^2}rdrd\theta$
Are the limits in the above integral true?
Best Answer
The domain is a half-disk of radius $a$, on the side of the positive $y$.
In Cartesian coordinates, $x$ runs from $-a$ to $a$ and $y$ is between $0$ and $\sqrt{a^2-x^2}$ (where $x^2+y^2=a^2$).
In polar coordinates, $\theta$ runs from $0$ to $\pi$ and $r$ from $0$ to $a$.
In both cases,
$$x^2+y^2\le a^2\land y\ge0.$$
Indeed $$x^2\le a^2\implies x^2+(a-x^2)\le a^2,$$
and
$$r^2\le a^2\land \theta\in[0,\pi]\implies\sin\theta\ge0.$$