Swirling vortex

$$ First, we obtain the intersection points $$ $\begin{aligned} 2y^2&=4y\\ y^2-2y&=0\\ y(y-2)&=0\\ y&=0 \,\text{ or }\, y=2. \end{aligned}$ $$ Thus, the volume is $$ $\begin{aligned} V&=\pi\int_0^2 \left((4y)^2-(2y^2)^2\right)\,dy\\ &=\pi\int_0^2 \left(16y^2-4y^4\right)\,dy\\ &=\pi\left.\left(\frac{16}{3}y^3-\frac{4}{5}y^5\right)\right|_0^2\\ &=\frac{256}{15}\pi\\ &\approx\boxed{53.6165} \end{aligned}$ $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$