Hello Kyle,
Please see http://mathhelpboards.com/math-notes-49/solid-revolution-about-oblique-axis-rotation-6683.html for the development of the general formula:
$$V=\frac{\pi}{\left(m^2+1 \right)^{\frac{3}{2}}}\int_{x_i}^{x_f} \left(f(x)-mx-b \right)^2\left(1+mf'(x) \right)\,dx$$
We first find $x_i$ and $x_f$:
$$x^2=x$$
$$x(x-1)=0$$
Hence:
$x_i=0,\,x_f=1$
We are given:
$m=1,\,b=0,\,f(x)=x^2\implies f'(x)=2x$
Hence:
$$V=\frac{\pi}{2\sqrt{2}}\int_{0}^{1} \left(x^2-x \right)^2(1+2x)\,dx$$
Expanding the integrand, we have:
$$V=\frac{\pi}{2\sqrt{2}}\int_{0}^{1} 2x^5-3x^4+x^2 \,dx$$
Applying the FTOC, we find:
$$V=\frac{\pi}{2\sqrt{2}}\left[\frac{1}{3}x^6-\frac{3}{5}x^5+\frac{1}{3}x^3 \right]_{0}^{1}$$
$$V=\frac{\pi}{2\sqrt{2}}\left(\frac{1}{3}-\frac{3}{5}+\frac{1}{3} \right)=\frac{\pi}{2\sqrt{2}}\left(\frac{1}{15} \right)=\frac{\pi}{30\sqrt{2}}$$