Looking down the 'throat' of the cylinder along the axis, we have something that looks kind of like this.
From the diagram,
But x=r. So,
I see what they done.
They broke the drilled hole up into octants(hence the multiplication by 8), then multiplied the area of the base by the height.
The height being
and the area of the base being
Of course, then they integrated over the region that makes up the hole.
That particular integral is a booger. I can see why they said do not evaluate.
Actually, thinking about it, what I worked out is for a hole through a sphere instead of a cylinder. It gives the volume remaining after a hole is drilled through a sphere. You can keep it if you wish or I can delete it.
Okay I took the x section to be a rectangle. The width of the rectangle
is a piece of a circle with respect to y so it is sqrt(R^2-y^2) with R being
the radius of the cylinder. The length of the rectangle is the y of the
piece of the hole (which is also circular) with the equation of sqrt (r^2-y^2).
The area of the rectangle varies along the radius of the holes so
the two circular functions are put in l*w form. What confuses me is why the
8 is there.