# What is the surface area produced by rotating f(x)=x^3-8, x in [0,2] around the x-axis?

Jun 21, 2018

first find dS, note the radius of the rotation is x.
$\mathrm{dS} = \sqrt{1 + 9 {x}^{4}} \mathrm{dx}$
${\int}_{0}^{2} 2 \pi x \mathrm{dS}$
to solve make a substitution of $w = 3 {x}^{2}$

#### Explanation:

If you make the substitution with w your integral becomes
${\int}_{0}^{12} \frac{\pi}{3} \cdot \sqrt{1 + {w}^{2}} \mathrm{dw}$
with a trigonometric substitution $w = \tan \left(\theta\right)$
the integral becomes
$\frac{\pi}{3} {\int}_{0}^{\arctan} \left(12\right) {\sec}^{3} \left(\theta\right) d \left(\theta\right)$
which you can solve by parts or using a table.