Finding volume of a solid rotated about a given axis of rotation

Question

Find the volume of the solid obtained by rotating the region bound by the curves $x=y^4$ and $x=1$ about the line $x=3$.

Outer radius $= r_O$

Inner radius $= r_I$

I was using the formula:

$V=\int_0^1 \pi (r_O^2-r_I^2)\,dy$

I was using Inner radius $= 2$, Outer radius $= 3-y^4$.

The answer that I keep getting is $\pi(5 - {6 \over 5} + {1 \over9})$, but this is incorrect.

Any insight? Thanks so much in advance.

Answer 1

Hint: The curve $x=y^4$ has a part below the $x$-axis. Integrate from $y=-1$ to $y=1$, or exploit symmetry.