Spin That Shark Fin!

$\begin{cases} x=y^2-4\\x=3y\end{cases}$

$\therefore y^2-4=3y$

$y^2-3y-4=0$

$(y-4)(y+1)=0$

$y=4$ or $y=-1$

When $y=4,x=12$

When $y=-1,x=-3$

$y^2-4=x$

When $x=0$ ,

$y^2-4=0$

$y^2=4$

$y=2$ or $y=-2$

We denote that the area of $\textcolor{#20A900}{green}$ region is the reflection of the $\textcolor{#3D99F6}{blue}$ region in the second quadrant.

Notice that the $\textcolor{#69047E}{purple}$ region is the overlapping region between the $\textcolor{#3D99F6}{blue}$ and $\textcolor{#20A900}{green}$ region in the first quadrant.

Note that by symmetry, the area of $\textcolor{#D61F06}{red}$ region $=$ the area of $\textcolor{#69047E}{purple}$ region

Net volume of solid generated

$=$ Volume of solid generated by rotating the $\textcolor{#3D99F6}{blue}$ region in first quadrant about y-axis

$+$ Volume of solid generated by rotating the $\textcolor{#20A900}{green}$ region in first quadrant about y-axis

$-$ Volume of solid generated by rotating the $\textcolor{#69047E}{purple}$ region in first quadrant about y-axis

$=$ Volume of solid generated by rotating the $\textcolor{#3D99F6}{blue}$ region in first quadrant about y-axis

$+$ Volume of solid generated by rotating the $\textcolor{#20A900}{green}$ region in first quadrant about y-axis

$-$ Volume of solid generated by rotating the $\textcolor{#D61F06}{red}$ region in second quadrant about y-axis

Notice that the volume of solid generated by the $\Delta BCD$ is a cone with radius $12$ and height $4$ .

Volume of solid generated by rotating the $\textcolor{#3D99F6}{blue}$ region in first quadrant about y-axis

$=\cfrac{1}{3}\pi(12)^2\cdot4-\pi\int_{2}^{4}(y^2-4)^2dy$

$=192\pi-\pi\int_{2}^{4}(y^2-4)^2dy$

Notice that the volume of solid generated by the $\Delta DAE$ is a cone with radius $3$ and height $1$ .

Volume of solid generated by rotating the $\textcolor{#20A900}{green}$ region in first quadrant about y-axis

$=\pi\int_{-2}^{1}(y^2-4)^2dy-\cfrac{1}{3}\pi(3)^2\cdot1$

$=\pi\int_{-2}^{1}(y^2-4)^2dy-3\pi$

Volume of solid generated by rotating the $\textcolor{#D61F06}{red}$ region in second quadrant about y-axis

$=\pi\int_{-2}^{-1}(y^2-4)^2dy+\cfrac{1}{3}\pi(3)^2\cdot1$

$=\pi\int_{1}^{2}(y^2-4)^2dy+3\pi$

Net volume of solid generated

$=192\pi-\pi\int_{2}^{4}(y^2-4)^2dy+\pi\int_{-2}^{1}(y^2-4)^2dy-3\pi-\pi\int_{1}^{2}(y^2-4)^2dy-3\pi$

$=186\pi-\pi(\int_{2}^{4}(y^2-4)^2dy+\int_{1}^{2}(y^2-4)^2dy)+\pi\int_{-2}^{1}(y^2-4)^2dy$

$=186\pi-\pi\int_{1}^{4}(y^2-4)^2dy+\pi\int_{-2}^{1}(y^2-4)^2dy$

$=186\pi-\pi\int_{1}^{4}(y^4-8y^2+16)dy+\pi\int_{-2}^{1}(y^4-8y^2+16)dy$

$=186\pi-\pi[\cfrac{y^5}{5}-\cfrac{8y^3}{3}+16y]_{1}^{4}+\pi[\cfrac{y^5}{5}-\cfrac{8y^3}{3}+16y]_{-2}^{1}$

$=186\pi-\pi(\cfrac{1024-1}{5}-\cfrac{8(64-1)}{3}+16(4-1))+\pi(\cfrac{1+32}{5}-\cfrac{8(1+8)}{3}+16(1+2))$

$=186\pi-\pi(\cfrac{1023}{5}-\cfrac{8(63)}{3}+16(3))+\pi(\cfrac{33}{5}-\cfrac{8(9)}{3}+16(3))$

$=186\pi-\pi(\cfrac{1023}{5}-\cfrac{8(63)}{3})+\pi(\cfrac{33}{5}-\cfrac{8(9)}{3})$

$=186\pi+\pi(-\cfrac{1023}{5}+\cfrac{8(63)}{3}+\cfrac{33}{5}-\cfrac{8(9)}{3})$

$=186\pi+\pi(-\cfrac{990}{5}+\cfrac{8(54)}{3})$

$=186\pi+\pi(-198+144)$

$=\pi(186-198+144)$

$=132\pi$

$\therefore V=\boxed{132}$