Solve Surface Integral for xy-Plane Projection

[awvvu]

Homework Statement

I'm not sure how to convert this surface integral into a double integral for evaluation.

$$\iint_S \frac{1}{1 + 4(x^2 + y^2)} dS$$

S is the portion of the paraboloid $z = x^2 + y^2$ between z = 0 and z = 1.

The Attempt at a Solution

How do you project this onto the xy-plane? From z = 0 to z = 1, we get circles of increasing radius. So do I just take z = 1, which is a circle of radius 1, and make the region:

$$D = \{(r, \theta) | 0 <= r <= 1, 0 <= \theta <= 2 \pi\}$$

Also, I know:

$$dS = \sqrt{1 + (2x)^2 + (2y)^2} dA = \sqrt{4r^2 + 1} dA$$

$$\iint_S \frac{1}{1 + 4(x^2 + y^2)} dS = \iint_D \frac{r}{1 + 4r^2} \sqrt{4r^2 + 1} dA$$

I just don't know what the limits should be. The region I chose gives me the right answer but I'm not sure exactly why the projection was when z = 1. What if it was the paraboloid between z = 1 and z = 2 then? Do I have to split up the surface integral to be the circle at z = 1 and the paraboloid surface? I'm really confused about this projection stuff.

 

[Dick]

The plane z=a cuts the paraboloid in the circle a=x^2+y^2. So the domain in the x,y plane is x^2+y^2<=a or r^2<=a. If the limits were z=1 to z=2, then the r limits would be r=1 to r=sqrt(2).