Here's the question from an exam that I couldn't solve:
If $\int_1^2 \int_0^x \frac{1}{(x^2+y^2)^\frac{3}{2}} ~\mathrm{dy} ~\mathrm{dx}$ transforms to $\int_0^a \int_b^c \frac{1}{r^2} ~\mathrm{dr} ~\mathrm{d\theta}$ in the polar coordinates $(r, \theta)$, then find $a, b, c$.
$a=\frac{\pi}{4}$, $b=\sec \theta$, $c=2\sec \theta$
the angle of the line $y=x$ is $\frac{\pi}{4}$ then you get for $r$ two triangles, with $r$ as hypotenuse,one with base $1$ and Another base $2$, where the angle is $\theta$. Unfortunately I dont know how to make pictures here.
The region is bounded left and right by the lines $x=1$ and $x=2$. If you draw a general radius of angle $\theta$ it will intersect these two lines forming triangles with $r$ as hypotenuese and Bases $1$ and $2$, now you can solve these triangles to find $r$ in terms of $\theta$.
