Using Symmetry for finding volume

Question

I have a confusion regarding the symmetry of the volume in the following question.

Find the volume common to the sphere $x^2+y^2+z^2=16$ and cylinder $x^2+y^2=4y$.

The author used polar coordinates $x=rcos\theta$ snd $y=rsin\theta$ and does something like this:

Required volume $V=4\int_0^{π/2}\int_0^{4sin\theta}(16-r^2)^{1/2}rdrd\theta$. The reason for multiplying by $4$ is the symmetry of the solid w.r.t. $xy$-plane.

My point of confusion is that this solid cannot be cut into $4$ identical parts, so how it can be multiplied by $4$?

Answer 1

My point of confusion is that this solid cannot be cut into 4 identical parts

Sure it can.

The sphere is centered at the origin, and the axis of the cylinder (which has radius $2$) intersects $(0,2)$ in the $x-y$ plane.

Slice it in half with the $z=0$ plane. Then slice it again with the $x=0$ plane.

These are reflected in the limits of integration.

Answer 2

Before that what happened to the latitude $\phi$ ?

The picture shows a situation Viviani suitable for what you described.

Two fold symmetry is evident as also factor 4.

Answer 3

A picture is useful here. The solid can indeed be split into four symmetric segments, as shown in the following surface plot.

Answer 4

The volume can be cut into 4 identical parts. See that both volume are simetric for each quadrant. That's why the angle variation is from $0$ to $\pi/2$.