Solve the Problem $z$ -Axis Shell: Upper Hemisphere Of and Density $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$

Multiple Choice

Solve the problem.
-The shape and density of a thin shell are indicated below. Find the moment of inertia about the $z$ -axis. Shell: upper hemisphere of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ cut by the plane $z = 0$ Density: $\delta = 4$

A) $\mathrm { I } _ { \mathrm { Z } } = 64 \pi$
B) $\mathrm { I } _ { Z } = \frac { 4096 } { 3 } \pi$
C) $\mathrm { I } _ { \mathrm { Z } } = 128 \pi$
D) $\mathrm { I } _ { \mathrm { Z } } = 512 \pi$