Calculating Volume using the Shell Method

Question

Hey I'm trying to do this problem by taking the integral from $0$ to $1$ of $2\pi{x}(x^4-(x^{1/4})dx$ and my answer comes out to $2\pi(\dfrac{1}{6} - \dfrac{4}{9})$ but that answer is incorrect. Can anyone help?

Answer 1

On the interval $[0,1]$, $x^{1/4}\geq x^4$. Hence the shell method yields: $$V=2\pi\int^1_0x(x^\frac{1}{4}-x^4)dx\\ \implies V=2\pi\left[\frac{4x^{\frac{9}{4}}}{9}-\frac{x^6}{6}\right]^1_0$$