# 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?

Jun 21, 2016

not including the area of the end caps, S = $540 \setminus \sqrt{2} \setminus \setminus \pi$, for the end caps add in $\pi \left({6}^{2} + {18}^{2}\right)$ also

#### Explanation:

if you look at the parameterisation, $x = y = {t}^{3} - t$ so this is a straight line.

and the interval specified corresponds in Cartesian to $\left(6 , 6\right) \setminus \to \left(24 , 24\right)$

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 ${\mathrm{ds}}^{2} = {\mathrm{dx}}^{2} + {\mathrm{dy}}^{2}$ we can say that ${\left(\setminus \frac{\mathrm{ds}}{\mathrm{dt}}\right)}^{2} = {\left(\setminus \frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\setminus \frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}$ and so $\mathrm{ds} = \setminus \sqrt{{\left(\setminus \frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\setminus \frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$

the elemental surface area of revolution is therefore $\mathrm{dS} = 2 \setminus \pi \setminus y \setminus \mathrm{ds}$

so Surface Area is $S = 2 \setminus \pi \setminus \int y \left(t\right) \setminus \sqrt{{\left(\setminus \frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\setminus \frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \setminus \mathrm{dt}$

and as the params are the same, $\setminus \frac{\mathrm{dy}}{\mathrm{dt}} = \setminus \frac{\mathrm{dx}}{\mathrm{dt}} = 3 {t}^{2} - 1$

so we have

$S = \setminus \sqrt{2} \setminus 2 \setminus \pi \setminus {\int}_{t = 2}^{3} \setminus \left({t}^{3} - t\right) \left(3 {t}^{2} - 1\right) \setminus \mathrm{dt}$

$= 540 \setminus \sqrt{2} \setminus \setminus \pi$

This is what you would get from the formula for the surface are of a truncated cone $S = \pi \left(s \left(R + r\right)\right)$ where s is slant height $s = \setminus \sqrt{{\left(R - r\right)}^{2} + {h}^{2}}$

NB this does not include the area of the end caps.