So I have this "double" integral:
$$\int_{ \frac{\pi}{4}}^{\frac{\pi}{2}} d{\phi} \int_{ \frac{cos(\phi)}{{sin}^2\phi}}^{\sqrt2} \frac{1}{3} r^3 sin(2\phi) d{r} +\int_{ \frac{5\pi}{4}}^{\frac{3\pi}{2}} d{\phi} \int_{ -\frac{cos(\phi)}{{sin}^2\phi}}^{\sqrt2} \frac{1}{2} r^3 sin(2\phi) d{r} $$
It is given in polar coordinates. So i have to define integration region and sketch it, i know i can check it with wolfram. This sum of integrals seems like i can somehow make it into one, since it's simmilar, at least boundaries are, but not sure.
After that i have to convert into cartesian coordinates. I am wondering how to explicitly define this integration region, since i have a sum. Any tips or help would be appreciated. I am used to solving from cartesian into polar, not the other way around, so this is a bit of a tougher challenge.
I will be grateful for every tip or answer, so thank you in advance.
Take out the constant terms (1/3 and 1/2). If you examine the graphs of $sin^2(\phi)\ cos(\phi)\ sin(2\phi)$ you will see that the two integrals are the same (note that the cos is of opposite sign in the two domains, compensated by the change of sign of the r integral lower limit). Combine them with the sum of the coefficients.