Use the parametrisation given inFig. Surface of a torusand calculate the volume of a torus!
Reminder
According to the Fig. Surface of a torus , the parametrization of half a circle with radius r, with centre at the x axis at x=R is given by
with R>r>0 and the interval of the parameter θ : [0,π].
A continuous single valued function y(x) in an interval of x [a,b] , where , defines a surface in the (x,y) plane between the x axis and the curve y(x) , and is bounded by the interval of x . When rotated by an angle of 2π about the y axis, the surface covers a volume of revolution.
(2.2.4.13)
Parts
The volume of revolution for the torus should be multiplied by 2, since we use only half a circle, which yields according to (2.2.4.13)