Answer
$\dfrac{7 \pi}{5}$
Work Step by Step
The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$
Now, $V=2\pi \int_{0}^{1} (2+x) (x^3) \ dx\\= 2\pi \int_{0}^{1} (2x^3+x^4) \ dx \\=2 \pi [\dfrac{ x^4}{2} +\dfrac{x^5}{5}]_{0}^{1} \\= 2\pi (\dfrac{7}{10}) \\=\dfrac{7 \pi}{5}$
