so I have to evaluate the integral
$$\int\int xy^2 \,dx\,dy$$ where the area is bound between two concentric circles:
$x^2 + y^2 = 2$ and $x^2 + y^2 = 4$
After introducing polar coordinates, I get that:
$ r^2=2$
$ r^2 = 4$
$ \phi \in [0, 2\pi]$.
$\phi$ is bounded between these boundaries because $r$ is not a function of $\phi$ so we take the "full" angle.
Now, I have a few questions. What are the boundaries for $r$? I chose them to be $ r \in [\sqrt{2}, 2]$ but I doubt that's correct. However, whichever bounds for $r$ I use, I get that the integral evaluates to zero, because:
$$ \int_0^{2\pi}\int_{\sqrt{2}}^2r^3\cos(\phi)\sin^2(\phi) \, dr \, d\phi$$
The integral of $\cos(\phi)\sin^2(\phi)$ for these bounds evaluates to zero, which makes me think I chose the wrong boundaries for the angle.
Can anyone help?
The integrand is an odd function of $x$ and the region has symmetry across the $x=0$ line thus
$$I = \iint_D xy^2\:dA = -\iint_D xy^2\:dA = -I \implies I=0$$