How do you find the volume using polar coordinates?

Question

The solid bounded by x^2+y^2=1, z=sqt(x^2+y^2), z=0.
Represent a cylinder extending along the z- axis and a cone above the z-axis, respectively.
Any ideas??
Thanks in advance!

Answer 1

You have a cylinder with radius 1 centered on the z-axis with one base the plane z = 0 and the other base at z = 1 (Think about it. This is where the cone has the same radius as the cylinder and so provides a top for the enclosed volume). Out of that you need to subtract a cone z = √(x² + y²).

You have a cylinder with radius 1 and height 1. You also have a cone with base radius 1 and height 1. Since the cone has 1/3 the volume of the cylinder, the answer should be 2/3 the volume of the cylinder.

Calculate the two volumes separately and then subtract. It will be easy to check the answer since, if you wanted to, you could work it up using nothing more than high school geometry.