# Applications of the Definite Integral- Volume

1. Sep 21, 2006

### ashleyk

Hi, I am having trouble visualizing this problem, if anyone can help me see it, I know how to do the integral part.

A church steeple is 30 feet tall with square cross sections. The square at the base has side 3 feet, the square at the top has side 6 inches, and the sides varies linearly in between. Compute the volume.

I assume the 3 feet and then 6 inches has to be in the same form so it would 36 inches and 6 inches. I also know that volume for a square would be length times width times height or one side raised to the 3rd power. I am just not sure to start.

2. Sep 21, 2006

### StatusX

A general fact about "pyramid" shapes, ie, shapes formed by starting with a base of a certain (2D) shape and sweeping out a volume by translating the shape and simultaneously linearly shrinking it to a point, is that their volume is 1/3 A*h, where A is the area of the base and h is the distance from the plane of the base to the tip. To prove this, note that as the side lengths linearly shrink, the area of the cross sections shrink quadratically, and integrating this out gives the result. In this problem, just take the difference of two such shapes.