Where does the formula for the volume of a hemisphere come from?

Question

Where does the formula for the volume of a hemisphere come from, I have been using this The formula

Answer 1

What I'm wondering, I have never seen this formula for spheres? What does $h$ mean? Anyway here's a calculation for the Volume of a sphere with radius $R$:

$$ \iiint_{\|\vec{x}\|=R}\mathrm{d}x\mathrm{d}y\mathrm{d}z\\ =\int_{\varphi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^R\sin(\theta)*r^2\mathrm{d}r\mathrm{d}\theta\mathrm{d}\varphi\\ =2\pi*\int_{\theta=0}^\pi\sin(\theta)\int_{r=0}^Rr^2\mathrm{d}r\mathrm{d}\theta\\ =2\pi*\frac{R^3}{3}\int_{\theta=0}^\pi\sin(\theta)=\frac{4\pi R^3}{3} $$

If we want a hemisphere: $$ \frac{1}{2}*\frac{4\pi R^3}{3}=\frac{2\pi R^3}{3} $$