# What is the surface area produced by rotating f(x)=x^3-x^2+1, x in [0,3] around the x-axis?

Feb 15, 2017

$\frac{9837}{70} \pi = 441.5$ cubic units, nearly.

#### Explanation:

graph{(x^3-x^2+1-y)(x-3+.001y)y(x-.001y)=0 [0, 4, -1, 20]}

Volume =$\pi \int {y}^{2} \mathrm{dx}$, with $y = {x}^{3} - {x}^{2} + 1$and x from 0 to 3

$= \pi \int {\left({x}^{3} - {x}^{2} + 1\right)}^{2} \mathrm{dx}$, with x from 0 t0 3

$= \pi \int \left({x}^{6} + {x}^{4} + 1 - 2 {x}^{5} - 2 {x}^{2} + 2 {x}^{3}\right) \mathrm{dx}$, for the limits

$= \pi \left[{x}^{7} / 7 + {x}^{5} / 5 + x - \frac{1}{3} {x}^{6} - \frac{2}{3} {x}^{3} + \frac{1}{2} {x}^{4}\right]$, between x = 0 and 3

$= \pi \left[\left(\frac{2187}{7} + \frac{243}{5} - 243 - 18 + \frac{81}{2}\right) - \left(0\right)\right]$

$= \frac{9837}{70} \pi$

#(140.529)(3.1416)=441.5 cubic units, nearly.