Rotate a below-axis region through a full turn

Factor $y=x^2(x-2)$. The bounded region lies between $x=0$ and $x=2$, below the x-axis. Four right angles mean $360^\circ=2\pi$ radians, a full turn. The source callout says to sketch first; the Teacher shades this exact region.

The radius is the distance to the axis, $|y|$, so its square is $y^2$. Therefore $V=\pi\int_0^2y^2\,dx=\pi\int_0^2(x^3-2x^2)^2\,dx$.

The Teacher gives $3.83$ cubic units to three significant figures. Independently, expanding gives $V=\pi[x^7/7-2x^6/3+4x^5/5]_0^2=128\pi/105\approx3.829751044$, agreeing with the source decimal.