MHB Integration in polar coordinates

[Fernando Revilla]

I quote a question from Yahoo! Answers

By changing to polar coordinates, evaluate the integral.
(Integrand)(integrand)[(x^2+y^2)^(7/2)… where D is the disk x^2+y^2<=16.

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

Denote $I=\displaystyle\iint_{D}(x^2+y^2)^{7/2}dxdy$ wiith $D\equiv x^2+y^2\le 16$. We have: $D\equiv \left \{ \begin{matrix}0\le \theta\le 2\pi\\0\le \rho \le 4\end{matrix}\right.$, so $$I=\int_0^{2\pi}d\theta\int_0^4(\rho^2)^{7/2}\rho d\rho=2\pi \left[\frac{\rho^9}{9}\right]_0^4=\frac{2^{19}\pi}{9}$$