Answer
$\dfrac{16 \pi}{35} $
Work Step by Step
The volume of a revolution can be calculated as:
$V=\pi \int_{m}^{n} (R^2_{outside}-R^2_{inside}) \ dy$
where, $R_{outside}=y^{1/3} $ and $ R_{inside}=y^3$
Now, $V=\pi \int_0^1 (y^{1/3})^2 -(y^3)^2 \ dy =\pi \int_0^1 (y^{7/3} -y^6) \ dy =\pi [\dfrac{3 y^{5/3}}{5}-\dfrac{y^7}{7}]_0^1 \\=\dfrac{16 \pi}{35} $
