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.

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- Jan 6th 2011, 01:53 AMalexmahoneVolume of revolution
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. - Jan 6th 2011, 04:22 AMearboth
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: