Given a solid with circle base of radius 2 and cross sections which are equilateral triangles, compute the volume of the solid.

We may describe the top half of the circular base of the solid by the equation

Thus, the length of the base of any equilateral triangular cross section is

Since these are equilateral triangles with side length , the area is given by

Then we compute the volume,

(**Note:** Apostol gives the solution in the back of the book, but I keep getting , as does Edwin in the comments. I’m marking this as an error in the book for now. If you see where my solution is wrong and Apostol is correct please leave a comment and let me know.)