What i the volume of the solid
y=3x-x^2, x=3, y=0, revolved about the y-axis
Shell method
$\displaystyle {V_y} = 2\pi \int\limits_a^b {xydx} = 2\pi \int\limits_0^3 {x\left( {3x - {x^2}} \right)dx} = 2\pi \int\limits_0^3 {\left( {3{x^2} - {x^3}}\right)dx} =$
$\displaystyle = \left. {2\pi \left( {{x^3} - \frac{{{x^4}}}{4}} \right)} \right|_0^3 = 2\pi \left( {27 - \frac{{81}}{4}} \right) = 2\pi \cdot \frac{{27}}
{4} = \frac{{27}}{2}\pi .$