Question

Find the volume of the solid bounded by the sphere x^2+y^2+z^2=2z

Answer 1

`x^2+y^2+z^2=2z\implies x^2+y^2+z^2-2z+1=1\implies x^2+y^2+(z-1)^2=1`

is a sphere centered at (0, 0, 1) with radius 1. We only need to know the radius; the volume is


`V=\frac43\pi r^3`

where
`r=1`, so the volume would be
`\frac{4\pi}3`.