Find the volume of the solid obtained by rotating about the y-axis the region bounded by $\displaystyle {y}={4}{x}^{{2}}-{x}^{{3}}$ and $\displaystyle {y}={0}$.

(a)   A typical shell has radius $\displaystyle {x}$, circumference $\displaystyle {2}\pi{x}$,
and height $\displaystyle {f{{\left({x}\right)}}}=$ Preview Question 6 Part 1 of 4  

(b)   So, by the shell method, the volume is
$\displaystyle {V}={\int_{{0}}^{{a}}}$ $\displaystyle {\left.{d}{x}\right.}$ Preview Question 6 Part 2 of 4  
and $\displaystyle {a}=$

(c)   Evaluate the integral: $\displaystyle {V}=$ Preview Question 6 Part 4 of 4