Can I see from here if this integral is zero?

Question

I'm computing the following integral:

$$\int_0^{\pi/4}\int_0^{2\pi} \sin^2(\rho)\sin(\theta)-\cos(\rho)\sin^2(\rho)\sin(\theta)~d\theta d\rho$$

Till this point I did everything correct, since when I compute it with a calculator it gets $0$ which is also the right solution. But is there a way to say now that it gets zero? Because I can't see how to procede and maybe one can argue with odd functions?

Thanks for your help.

Answer 1

The function is odd with respect to the line $\theta = \pi$, so the integral is zero.

Answer 2

Write the integral as $$\left(\int_0^{\pi/4}\sin^2(\rho)-\cos(\rho)\sin^2(\rho)\,d\rho\right)\int_0^{2\pi}\sin\theta\,d\theta.$$ Since $\int_0^{2\pi}\sin(\theta)\,d\theta=0$ (show this directly or by symmetry about $\pi$), the entire expression is zero.