Double integral $\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y$ Solve the following double integral
\begin{equation}
\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y
\end{equation}
where $D: \{(x,y)\mid x^2+y^2\leq y \}$.
Some help please? Thank you very much.

This may be done easily in polar coordinates; the equation of the circle is $r=\sin{\theta}$, $\theta \in [0,\pi]$.  The integrand is $r^6 |\cos^3{\theta} \sin^3{\theta}|$, and is symmetric about $\theta = \pi/2$.  The integral is then
$$2 \int_0^{\pi/2} d\theta \, \cos^3{\theta} \, \sin^3{\theta}\, \int_0^{\sin{\theta}} dr \, r^7 = \frac14 \int_0^{\pi/2} d\theta\, \cos^3{\theta} \, \sin^{11}{\theta} $$
which may be evaluated simply as
$$ \frac14 \int_0^{\pi/2} d(\sin{\theta}) (1-\sin^2{\theta}) \sin^{11}{\theta} = \frac14 \left (\frac{1}{12} - \frac{1}{14} \right ) = \frac{1}{336} $$
