My questions pertains to why the following limits of integration would be the following given the following scenario:\begin{equation}\int_0^a \int_0^\sqrt{a^2-x^2} \frac{dy\ dx}{(1+x^2+y^2)^\frac{3}2} \end{equation} I have to convert it to polar coordinates and solve it, I got this as my as my region: \begin{equation}\int_0^\frac{\pi}2 \int_0^\sqrt{a} \frac{r}{(1+r^2)^\frac{3}{2}}dr d\theta \end{equation} But my region is supposed to be this: \begin{equation}\int_0^\frac{\pi}{2}\int_0^a \frac{r}{(1+r^2)^\frac{3}2}dr d\theta\end{equation}
Any thoughts to why is it particularly that?
You have
$$y = \sqrt{a^2 - x^2} \implies x^2 + y^2 = a^2 \tag{1}\label{eq1A}$$
This is the equation of a circle of radius $|a|$ centered at the origin. Since the original equation has $x$ going from $0$ to $a$, I'll assume that $a \gt 0$ so $|a| = a$. Thus, with $y$ going from $0$ to $\sqrt{a^2 - x^2}$ and $x$ going from $0$ to $a$, this would be the portion of the circle in the first quadrant. As such, $r$ would go from $0$ to $a$.