# Surface integral without using Gauss' theorem

1. Sep 20, 2009

### maupassant

1. The problem statement, all variables and given/known data

Calculate §§ A.n dS if
A= 2y(x^2)i-(y^2)j + 4xzk
over the region in the first octant bounded by (y^2)+(z^2) = 9 and x = 2

2. Relevant equations

3. The attempt at a solution

Let n = (yj + zk) / 3

then A.n = [-(y^3) +4xz^3] / 3

Since we 'll project the surface onto the xy-plane:
|n.k| = z/3 and z = SQRT(9-y^2)

Putting all together I obtain
= §§R (4xz^3 - (y^3))/z dx dy

Now making the appropriate changes and setting up the limits of integration:

§y=30 §x=20 4x(9-y^2) - (y^3)/sqrt(9-y^2) dx dy

However I always obtain 108 as a result and not 180 as my book suggested me (and after verification by Gauss' divergence theorem.

Is there a problem with the limits of integration? Wrong projection? I really have no clue ...
Thanks for the help!

2. Sep 20, 2009

### gabbagabbahey

You've only calculated the integral over one side/face of the surface....there are three more faces that make up the closed surface bounding the given region...you need to calculate the surface integral over all 3 of those as well.

3. Sep 20, 2009

### maupassant

Thanks a lot!
I finally got it (at least I hope so ;-) !