I'm working on this problem:
Calculate $$ \iint_D x y^3 \ln(x^2+y^2) dx dy$$ where $D$ is $\{ (x,y) \in \mathbb R^2 :9 < x^2+y^2 < 49, (x,y)>0 \}$.
What I need help with is understanding how to determine the upper and lower limit of the integrals. And is it easier to calculate using polar coordinates?
Transform into polar coordinates $(\rho,\theta)$: $$x=\rho \cdot \cos{\theta}$$ $$y=\rho \cdot \sin{\theta}$$ $$\rho^2=x^2+y^2$$
Then the integrand will be: $$x y^3 ln(x^2+y^2)=2 \rho ^4 \sin ^3(\theta ) \cos (\theta ) \log (\rho )$$
Now do the integration:
$$\int _0^{\frac{\pi }{2}}\int _3^72 \rho ^4 \sin ^3(\theta ) \cos (\theta ) \log (\rho )\cdot \rho d\rho d\theta=\frac{1}{36} (-58460-2187 \log (3)+352947 \log (7))$$