Volume of a region of a sphere which is seperated with a plane

Question

We have a sphere with the center coordinates $(0,0,0)$ and a radius $a$. If we separate this sphere with the plane $y= a/2$ what will be the volume of the region of the sphere between $y=a/2$ and $y=a$. I tried to use cylindrical coordinates but I think I could not succeed at that.

Answer 1

This is a solid of revolution with circular cross-sections. It will just be

$$ \int_{a/2}^a\pi x^2\,dy $$

with $x^2=a^2-y^2$.