Parameterizing surface in surface integral problem

[tasveerk]

Homework Statement

Find the area of the surface cut from the paraboloid z=2x²+2y² by the planes z=2 and z=8.

Homework Equations

Surface area of S= ∫∫ ||T_s×T_t|| ds dt

The Attempt at a Solution

What I am really having trouble doing in this problem (and in general) is parameterizing the surface in terms of s and t. I think for this surface an accurate parameterization is (s,t,√(2s²+2t²)) but I am not sure. Also, I wouldn't know how the planes cutting the paraboloid would come into play.

 

[Dick]

Why did you put the square root into (s,t,√(2s^2+2t^2)) for a starting point? The planes will define the (s,t) region you will integrate over.

 

[tasveerk]

Good question. I seem to have made an error. I was looking at the parameterization of a different paraboloid and must have gotten mixed up.

 

[Dick]

Good question. I seem to have made an error. I was looking at the parameterization of a different paraboloid and must have gotten mixed up.

Ok, so just use (s,t,2s^2+2t^2). Now if z=2, then that means 2s^2+2t^2=2. What kind of curve is that in the (s,t) plane?

 

[tasveerk]

A circle of radius 1. When z=8, the curve will be a circle of radius 2. Not sure where to go from here.

 

[Dick]

A circle of radius 1. When z=8, the curve will be a circle of radius 2. Not sure where to go from here.

Ok again. So you are going integrate between those two circles. Suggests you might think about polar coordinates. I'm not sure why you are not sure where to go. The next thing would be to work on what to integrate, right? Try working on the partial derivative vectors and finding the cross product.