Answer
$\dfrac{\pi}{3}$
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:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$
Now, $V=2\pi \int_{0}^{1} y (1-y) \ dy\\= 2\pi \int_{0}^{1} (y-y^2) \ dy \\=2 \pi [\dfrac{y^2}{2} -\dfrac{y^3}{3}]_{0}^{1} \\= 2\pi [\dfrac{1}{2}-\dfrac{1}{3}] \\=\dfrac{\pi}{3}$
