Question

Evaluate the following iterated integral by converting to polar coordinates.

Answer 1

Looks like the integral is

$\displaystyle\int_{-8}^8\int_0^{\sqrt{64-x^2}}\sin(x^2+y^2)\,\mathrm dy\,\mathrm dx$

The integration region is the upper half of a circle in the x-y plane centered at the origin with radius 8. So, in polar coordinates, the integral is

$\displaystyle\int_0^{2\pi}\int_0^8r\sin(r^2)\,\mathrm dr\,\mathrm d\theta=2\pi\int_0^8r\sin(r^2)\,\mathrm dr$

$=\pi\displaystyle\int_0^{64}\sin s\,\mathrm ds$

where we substituted $s=r^2$ and $\mathrm ds=2r\,\mathrm dr$, which gives

$=\boxed{\pi(1-\cos(64))}$