The principle to be used is:
If f(x) is an
even function, i.e. f(-x) = f(x), then if you rotate the bum-shaped region about y = b say, then:
![](https://latex.codecogs.com/png.latex?\bg_white V = 4 \pi b \int ^{a} _ 0 y dx = 4 \pi b A )
where 'A' is the area of half a bum (
part (b) is to show this).
You can now apply this principle to a circle radius 'r Bearing in mind, that the upper semi-circle, radius 'r' and centred at the origin. The upper semi-circle:
![](https://latex.codecogs.com/png.latex?\bg_white y = + \sqrt{r^2 -x^2} )
, which is an even function, if we rotate this semi-circle about x=s or x= -s, we get a volume of
![](https://latex.codecogs.com/png.latex?\bg_white 4 \pi s \times \frac{1}{4} \pi r^2 )
. When you apply same to lower semi-circle you get the same volume. Combining the two, you get the volume of the torus =
![](https://latex.codecogs.com/png.latex?\bg_white 2 \pi s \times \pi r^2 = 2 {\pi}^2r^2 s )
(??)