Answer
(a) $V=\frac{7 \pi}{30}$
(b) Easier
Work Step by Step
Write formule for the volume using cylindrical shells
(a) $\int_{0}^{1} 2 \pi x\left(x^{3}-3 x^{2}+2 x\right) d x=V$
Determine antiderivative
$$
\begin{array}{c}
\left.2 \pi\left(\frac{x^{5}}{5}-\frac{3 x^{4}}{4}+\frac{2 x^{3}}{3}\right)\right|_{0} ^{1}=V \\
\frac{7 \pi}{30}=V
\end{array}
$$
(b) The method of using cylindrical shells would be much simpler than the method of cutting, since in order for us to use the method of cutting, we need to decide the inverse function of the shell. $x^{3}-3 x^{2}+2 x=y$ (or rewrite this equation as $f(y))=x$ which will be very difficult to determine.
