Question

Derive the formula for the volume of a torus of internal radius \(r\) and outer radius \(R\) using the disk/washer method.

Answer 1

sorry, inner radius \(r\) is inside the hoop, as shown, and the outer \(R\) extends to the *center* of the smaller circle of radius \(r\)

Answer 2

pi*r^2*2*pi*R?

Answer 3

This is my jam. I love this.
\[2\int\limits_{R-r}^{R+r}2\pi x*\sqrt{r^2-x^2}dx\]

2 out front represents the top and bottom half. I have circumference at a point, height at a point from pythagoras, and infinitesimal width.

So how do you want me to proceed, I can solve the integral but that's sort of just downhill from here.

Answer 4

yes, but you could google the formula for a torus pretty quick. I'm looking for showing it with disk method

Answer 5

is mine correct?

Answer 6

@Kainui this is pretty good, but the bounds..,
@kc_kennylau yes

Answer 7

oh I see, sorry, you are doing it along x

Answer 8

bounds are fine

Answer 9

Yes, actually it's simple substitution. r^2-x^2=u

Answer 10

yeah, you are doing shell method, not disks, but that's okay @Kainui

Answer 11

Ahh, I thought either was acceptable, I can try the other way for fun I suppose haha.