Consider the tetrahedron with vertices at the origin and at the points where the plane
intersects the coordinate axes. Compute the volume of this tetrahedron.
First, the intercepts of the plane are given by . Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices is
Letting we have
Compute the volume of the parallelpiped determined by the vectors .
The volume is given by the scalar triple product
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.)