Find the volume of the solid formed by the revolving the region around a line

Question

I have to find the volume of the solid formed by the revolving the region enclosed by $x=\frac{y^2}4$ and $y=x^5+x^3$ around the line $y=2$

I know how to find the volume when it revolve around x-axis, but when it revolve around a function line or y-axis, how should I approach it? Appreciate anyone help

Answer 1

Transform $y$ into $y-2$, so that line $y=2$ is transformed into the $x$-axis. The two equations become $y=x^5+x^3-2$ and $y=2\sqrt x -2$.

Answer 2

Intersection of the planes that are perpendicular to the x-axis and the solid is the region(washer) with the area $$\pi[(2-x^5-x^3)^2-(2-2 \sqrt x)^2]$$. Therefore, $$V= \int_0^1 \pi[(2-x^5-x^3)^2-(2-2 \sqrt x)^2]dx$$