The volume subtracted out by drilling the hole is not precisely a cylinder, because a cylinder usually has flat ends. The material removed has curved ends (matches up with the sphere). So what do you get when you change this?
A hole of diameter d is drilled through a sphere of radius r in such a way that the axis of the hole passes through the centre of the sphere. Find the volume of the solid that remains.
I drew a picture of a sphere centred on the cartesian plane, with a hole going through its centre (in the shape of a cylinder).
I already knew that the volume of the sphere was . I then worked out the volume of the hole (cylinder):
Volume of solid remaining =
but the correct answer was
I cant see where I went wrong cause surely the objective of this problem is to minus the hole (cylinder) from a sphere to get the remaining volume?
Rather than subtracting the round-ended cylinder from the volume of the sphere, I think it's probably easier to set up the integral for the volume. You can use cylindrical shell elements of width , circumference , and height , and integrate from x = d/2 to x = R.