Take the tour ×  This is hard to explain but I'll do my best. I
hope I'm clear.  Imagine you have a donut. You want to find the
volume of it and the method you want to use is to imagine
slicing one side of that donut and opening it out into a
cylinder. Only, it's not exactly a cylinder as it has two
pointed ends: one side of the cylinder has the length of the
inner circumference of the donut, the other side has the length
of the outer circumference.  The 'middle part' is a simple
cylinder. I want to find the volume of each end part.  There's
an easy trick to it, but that's not the solution I'm looking
for; the easy trick being putting the two end parts together to
make a smaller cylinder.  But, how I want to do this is to find
the volume of one of these pointy cylindrical endparts with an
integral. However, I can't seem to hit the right answer.  Let's
say my stretched out donut has a left side of length   and a
right side of length  $2\pi(R+r)$ and a radius of $r$. Let's say
I slice the top and bottom pointed