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Compute the volume of a solid with given properties

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.)

Calculate the values of an integral of a step function

For define a step function on the interval by Then, define 1. Calculate .
2. Find all values of such that .

1. We calculuate: 2. .

Note: There is an error in the book. The answers in the back of the book claim that , which is incorrect.