Answer
a) $\dfrac{24 \pi}{5}$
b) $\dfrac{48 \pi}{5}$
Work Step by Step
a) $V= \pi \int_{0}^{1} (\sqrt x)^2-(\dfrac{x^2}{8})^2 dx=\dfrac{-x^2 \pi(x^3-160)}{320}=\dfrac{24 \pi}{5}$
b) We need to use the shell model as follows:
$V=\int_p^{q} (2 \pi) \cdot (\space radius \space of \space shell) ( height \space of \space Shell) \space dx \\= \int_{0}^{4} (2 \pi) \cdot (x) (\sqrt x-\dfrac{x^2}{8}) dx \\=\dfrac{-x^{5/2} \pi(5x^{3/2}-64)}{320} \\=\dfrac{48 \pi}{5}$
