Thread: Find the volume of the described solid.

A pyramid with height h and base an equilateral triangle with side a (a tetrahedron).

This is an example problem from the text. If someone could walk me through how to do this problem using integration, I think it will help me grasp how to do the other problems.

I'm having trouble starting, and not sure what I'm supposed to do. I think I'm supposed to find the surface area of the triangle with respect to height, and evaluate it from 0 to h. But how to I find out what the triangular base is at any given height?

Imagine the center of the equilateral triangle base sitting on the origin in the xy-plane. The figure comes to a point somewhere along the z-axis at the point (0,0,h).

The general formula for finding volume of such a figure is , where is the cross-sectional area of the figure at a distance r from the origin. Here, your bounds of integration will be . since it comes to a point, and since this is the area of an equilateral triangle with side length .

Since the side-length of the equilateral cross-sectional triangle shrinks linearly from a to 0, it can be represented as , so the area

And therein lies your integral:

(You can of course check your work by using the simple geometric formula for figures that start at a base of area b and rise to a point at height h.)

This is exactly what I needed to hear. It all makes sense now. I only wish I had gotten the reply before I had to go into class an hour and a half ago. Oh well, at least I understand it better now.