Answer
(a) $$\dfrac{24 \pi}{5}$$
(b) $$\dfrac{48 \pi}{5}$$
Work Step by Step
(a) $$Volume= (\pi) \int_{0}^{1} (\sqrt x)^2-(\dfrac{x^2}{8})^2 dx \\=\dfrac{-x^2 \pi(x^3-160)}{320}\\=\dfrac{24 \pi}{5}$$
(b) Consider the shell model to compute the volume:
$$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\=
\int_{0}^{4} (2 \pi) \cdot (x) \times (\sqrt x-\dfrac{x^2}{8}) dx \\=-\pi\times \dfrac{x^{(5/2)} (5x^{3/2}-64)}{320} \\=\dfrac{48 \pi}{5}$$
