Calculate the moment of inertia of a solid uniform hemisphere $x^2+y^2+z^2=a^2$; $z \geq 0$ with mass $m$ about the $z$ – axis.
My attempt: We know that MI of a solid hemisphere is $2Ma^2/5$ where $M$ is mass and a is radius of solid hemisphere. And the MI of a solid hemisphere is $Ma^2/5$. So in above question mass is given $m$ and radius is a so $MI$ of hemisphere is $2ma^2/5$. Is this correct answer ? Answer given is $15mπa/32$.
Assuming that the mass is m and the radius a, you do not need to integrate in order to find out the MI of a solid hemisphere.
$$MI_{sphere}=\frac{2Ma^2}{5}$$
In the process of making a hemisphere out of it we know that we are eliminating mass symmetrically from the axis, i.e.
$$MI_{cut out}=MI_{hemisphere\ left\ out}$$
and their sum is $\frac{2Ma^2}{5}$.
Thus $$MI_{hemisphere}=\frac{Ma^2}{5}$$.
But remember that M is not the mass of your hemisphere.
$$M=2m$$ where m is the mass of your hemisphere and M that of whole sphere.
Thus $$MI_{hemisphere}=\frac{2ma^2}{5}$$