Thread: Double Integral with Polar Coordinates

Problem:

Use polar coordinates to find the volume of the given solid.

Enclosed by the hyperboloid

$\displaystyle -x^{2} - y^{2} + z^{2} = 1$ and the plane $\displaystyle z=2$

My attempt:

I think I'm tripping up on the radius portion of this problem. I tried to solve the function for z and set the two bounds equal to find their intersection and therefore the radius. I substituted z for two while finding the radius, which is something I probably should not do. Any help would be great. Attached is an image of my attempt.

Hi

The first thing to do is to draw a sketch.
The plane z=2 is easy to draw.
The hyperboloid is x²+y²=z²-1. Now x²+y² being positive, z is higher than 1 (or lower than -1 which we do not consider here). For any value k of z (higher than 1) the intersection of the plane z=k with the hyperboloid is the circle x²+y²=k²-1. And in polar coordinates x²+y²=r².

The volume is therefore $\displaystyle \int \int \int r dr d\theta dz$. I let you find the boundaries for each variable.