Compute the moment of inertia about the z-axis of the following objects with density, delta.
(a). The sphere $x^2+y^2+z^2\le a^2$
(b). The cylinder $x^2+y^2\le a^2$, $0\le z\le h$.
My attempt:
(a). Mass= $\frac{4}{3}\pi r^3$($\delta$)
radius of disk: $y=(a^2-x^2)^\frac{1}{2}$
volume of disk: $\pi(a^2-x^2)dx$
mass=(Density)(volume)=$(\delta)\pi(a^2-x^2)dx$ moment of inertia=[(mass of disk)(radius)^2]/2
[integral]: $[-a,a]$ $I=[\dfrac{(\pi)(\delta)}{2}][(a^2-x^2)^2dx]$
(kinda stuck here... not sure if this is right or in the right direction)
TEXTBOOK ANSWER: $\dfrac{2}{5}a^2(\delta\dfrac{4}{3}(\pi)a^3)=(\dfrac{2}{5}a^2)$(mass of sphere)
(b). radius=$a$
height=$h$, $0\le z\le h$ [integral]: $[0,M]$ I=[(dM)(a^2)/2] =[(a^2/2)(M)] =((a^2)/2)((pi)(a^2)(h)(delta))
\begin{equation}\begin{aligned} I&=[(dM)(a^2)]/2 \\ &= \dfrac{a^2}{2}M \\ &= (\dfrac{a^2}{2})(\pi a^2h)(\delta) \end{aligned}\end{equation} (again... I'm not confident that this is correct)
TEXTBOOK ANSWER: $\dfrac{a^2}{2}((\delta)(\pi)(a^2)(h))=(1/2)(a^2)$(mass of cylinder)