Hello, Jones!
$\displaystyle R$ is the region bounded by $\displaystyle x\,=\,y,\;x\,=\,4yy^2$ is rotated about the x axis.
Determine the volume of the solid. Code:

* *
 * *
 o(3,3)
 *::::*
 *::::::::o(4,2)
 *::::::::*
 *:::::::*
 *:::::*
  o              

Intersection: .$\displaystyle y \:=\:4y  y^2 \quad\Rightarrow\quad y^23y\:=\:0 \quad\Rightarrow\quad y(y3)\:=\:0$
. . The graphs intersect at (0,0) and (3,3).
"Shells" Formula: .$\displaystyle V \;=\;2\pi\int^b_a y\left(x_{\text{right}}  x_{\text{left}}\right)\,dy $
We have: .$\displaystyle V \;=\;2\pi\int^3_0y\bigg[(4yy^2)  y\bigg]\,dy \;=\;2\pi\int^3_0\bigg[3y^2  y^3\bigg]\,dy $
Now finish up . . .