Answer
The mass of a cylinder is $1280\pi $.
Work Step by Step
Let the cylinder be located on the $xy$-plane and its central axis be the $z$-axis. Since the mass density $\rho $ at a point is equal to the square of the distance from the cylinder's central axis, we have
$\rho \left( {x,y,z} \right) = {x^2} + {y^2}$
In cylindrical coordinates:
$\rho \left( {r\cos \theta ,r\sin \theta ,z} \right) = {r^2}$
Let ${\cal W}$ denote the region. So, the description of ${\cal W}$:
${\cal W} = \left\{ {\left( {r,\theta ,z} \right)|0 \le r \le 4,0 \le \theta \le 2\pi ,0 \le z \le 10} \right\}$
Compute the mass:
mass $ = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} \rho \left( {x,y,z} \right){\rm{d}}V$
$ = \mathop \smallint \limits_{\theta = 0}^{2\pi } \mathop \smallint \limits_{r = 0}^4 \mathop \smallint \limits_{z = 0}^{10} {r^3}{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $
$ = \left( {\mathop \smallint \limits_{\theta = 0}^{2\pi } {\rm{d}}\theta } \right)\left( {\mathop \smallint \limits_{r = 0}^4 {r^3}{\rm{d}}r} \right)\left( {\mathop \smallint \limits_{z = 0}^{10} {\rm{d}}z} \right)$
$ = 5\pi \left( {{r^4}|_0^4} \right) = 1280\pi $
So, the mass of a cylinder is $1280\pi $.
