Solve the problem.
-A solid right circular cylinder is bounded by the cylinder $r = 2$ and the planes $z = 0$ and $z = 3$. Find the moment of inertia about the $z$-axis if the density is $\delta = 2 \mathrm { z }$.
A) $36 \pi$
B) $72 \pi$
C) $48 \pi$
D) $144 \pi$

Question

Quizplus · Accepted Answer

Since the density is a function of only $z$, we can use cylindrical coordinates to evaluate the moment of inertia:
$$ I = \iiint_V \delta r^2 dV $$
Where $V$ is the volume of the cylinder. Since the cylinder is symmetric about the $z$-axis, we can use symmetry to simplify the above integral:
$$ I = 2 \iiint_V \delta r^2 dV $$
We can evaluate this integral using cylindrical coordinates:
$$ I = 2 \int_0^{2 \pi} \int_0^2 \int_0^3 2z (r^2) r dz dr d \theta $$
$$ I = 4 \pi \int_0^2 r^3 \left[ \frac{3^2}{2} \right] dr $$
$$ I = 54 \pi \int_0^2 r^3 dr $$
$$ I = 54 \pi \left[ \frac{1}{4} r^4 \right]_0^2 $$
$$ I = \boxed{72 \pi} $$

Solve the Problem r=2r = 2r=2 And the Planes z=0z = 0z=0

Solve the Problem $r = 2$ And the Planes $z = 0$