A cube C of edge length 1 is rotated around a line passing through two opposite vertices, thereby sweeping out a solid S of revolution. Find the volume of S.
Any hints would be much appreciated.
1. Draw a sketch. Basically you are dealing with a rectangle whose length is and whose width is .
Split this rectangle into two congruent right triangles.
2. By rotating the rectangle about it's diaogonal you'll get a solid composed of two cones and two frustrums of cones.
The height of the solid is .
The base radius of the two cones is:
The height of the two cones is:
The second radius of the two frustrums of cones is:
The height of the two frustrums of cones is: .
3. To calculate the complete volume of the solid use the formulas of the volume of a cone:
and the formula of the volume of a drustrum of a cone: