Answer
$\dfrac{11 \pi }{21}$
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, $V=2\pi \int_{0}^{1} x(x-x^{12}) \ dx \\ = 2\pi \int_0^1 [x^2-x^{13}] \ dx \\= 2 \pi [ \dfrac{x^{3}}{3}-\dfrac{x^{14}}{14}]_0^1\\=2 \pi ( \dfrac{1}{3}-\dfrac{1}{14}) \\= \dfrac{11 \pi }{21}$
