1. A group of mathematics professors have decided to build a temple to their god by building a conical pyramid. The cone, pictured at the right, will have the form generated by rotating the line y = 200 -x, 0<= x <= 200 around the y-axis.

The variables x any y are measured in feet.

The work to be done consists of hauling stone, which weighs 80 lbs a cubic foot,from ground level to its place in the pyramid. All the work done is against gravity, i.e., lifting the stone vertically; horizontal movement is not a factor in this problem.

 

 

 

 

 

 

The work in building the pyramid is to be done by student volunteers. What is the total amount of work they will do?

Solution:

Think of the pyramid as a set of circular disks whose radius at height y is x and volume is p x2 dy. Then the work done in lifting the blocks to the height y is

Consequently, the total work done is equal to the integral