Let R be the region bounded by $y=x^3$ and $y=x$ in the first quadrant. Find the volume of the solid generated by revolving R about the line $x=-1$
The region goes from $y=0$ to $y=1$. For an arbitrary $y$-value, say, $y=c$, $0\le c\le1$, what is the cross section of the region at height $c$? That is, what is the intersection of the region with the horizontal line $y=c$? What do you get when you rotate that cross-section around the line $x=-1$? Can you find the area of what you get when you rotate that cross-section? If you can, then you can integrate that area from 0 to 1 to get the volume.