Question

Find the volume of the solid formed by revolving the region bounded by the graph $\text{f}\left(x\right)$ = $-x^2$ +$\text{x}$ and the x-axis about the x-axis.

Answer 1

$Volume={\int }_0^1\pi y^{2}dx$

$\pi {\int }_0^1(-x^2+x{)}^{2}dx$

$=\pi [\frac{x^5}{5}-\frac{x^4}{2}+\frac{x^3}{3}{]}_0^1$

$=\pi [\frac{1}{5}-\frac{1}{2}+\frac{1}{3}]$

$=\pi [\frac{1}{5}-\frac{1}{6}]$

$=\frac{\pi }{30}=0.1043$