What is the surface area of the solid created by revolving #f(t) = ( t^3-t, t^3-t, t in [2,3]# around the x-axis?
not including the area of the end caps, S =
if you look at the parameterisation,
and the interval specified corresponds in Cartesian to
maybe there is a typo here as bothering with the calculus seems overkill but we can proceed anyways.
starting with the simple idea of arc length, viz that
the elemental surface area of revolution is therefore
so Surface Area is
and as the params are the same,
so we have
This is what you would get from the formula for the surface are of a truncated cone
NB this does not include the area of the end caps.