Using a triple integral to find volume of a hemisphere radius 1

Question

My question asks me to use a triple integral to prove that the volume of a hemisphere with radius 1 is $V=\frac{2}{3}\pi r^3$

I guess I should be using spherical coordinates but I am struggling to know where to start.

thanks universe

Answer 1

The volume element in spherical coordinates is $dV=r^2\sin \theta dr d\theta d \varphi$ And, for a hemisphere of radius $r=1$ the limits of integration are:

$$0<r \le 1 \qquad 0< \theta \le \frac{\pi}{2} \qquad 0<\varphi \le 2\pi $$

So we can calculate the volume with the triple integral: $$\int_V dV=\int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^1 r^2\sin \theta dr d\theta d\varphi $$

can you do from this?