I read an article not too long ago that posed the following problem:
What is the volume of the solid of revolution created by spinning a unit cube about an axis joining two opposing vertices?
So the shape generated will be two cones and a parabola-like curve in the "middle". I hope that makes sense. At first, I tried to find a cross section of the resulting structure and then integrating it with disks, but I think I am over-complicating it. How would I go about solving this problem if I define my unit cube to be on the first octant with $i, j,$ and $k$ (so the axis would be $r(t)=t\\langle1,1,1\\rangle,0 < t <1$)? Thank you.
If you draw and label a diagram you shall see that the rotating figure creates two cones each with a base diameter of square root 3 and a height of one-half the square root of 3. Now use the formula for volume of a cube.
