Answer
$$\dfrac{ 16\pi}{3}$$
Work Step by Step
Consider the shell model to compute the volume:
$$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = \int_{0}^{2} (2 \pi) \cdot y (2+y-y^2) dy \\= (2\pi) [y^2+\dfrac{y^3}{3}-\dfrac{y^4}{4}]_{0}^{2} \\=[4(2 \pi)+\dfrac{8(2 \pi)}{3}-4(2 \pi)] \\=\dfrac{ 16\pi}{3}$$
