A spherical shell centered at the origin has an inner radius of 3 cm and an outer radius of 5 cm. The density, $\delta$, of the material increases linearly with the distance from the center. At the inner surface, $\delta=8$ g/cm${}^3$; at the outer surface, $\delta=14$ g/cm$^3$.

(a) Using spherical coordinates, write the density, $\delta$, as a function of radius, $\rho$. (Type rho for $\rho$.)
$\delta =$

(b) Write an integral in spherical coordinates giving the mass of the shell (for this representation, do not reduce the domain of the integral by using symmetry; type phi and theta for $\phi$ and $\theta$).
With $a =$, $b =$,
$c =$, $d =$,
$e =$, and $f =$,
Mass = $\int_a^b\int_c^d\int_e^f$ $d$ $d$ $d$

(c) Find the mass of the shell.
Mass =
(Include units.)