Answer
$$\dfrac{ 8\pi}{3}$$
Work Step by Step
Consider the shell model to compute the volume:
$$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ =2 \pi \times \int_{0}^{2} y \times (y-\dfrac{y}{2}) \space dy \\= 2\pi [\dfrac{y^3}{6}]_{0}^{2} \\=2 \pi (\dfrac{8}{6}) \\=\dfrac{ 8\pi}{3}$$