Answer
$$\dfrac{\pi}{6}$$
Work Step by Step
Consider the shell model to compute the volume:
$$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = \int_{0}^{1} (2 \pi) \times y \times (2y-y^2-y) dy \\ =2 \pi \times (\dfrac{1}{3}-\dfrac{1}{4}) \\=\dfrac{\pi}{6}$$
