- #1

MacLaddy

Gold Member

- 291

- 11

## Homework Statement

Evaluate the surface integral ∫∫f(x,y,z)dS using an explicit representation of the surface.

[itex]f(x,y,z) = x^2 + y^2;\mbox{ S is the paraboloid } z= x^2 + y^2\mbox{ for }0\leq z \leq 4[/itex]

## Homework Equations

[itex]\displaystyle \int \int_{S} f(x,y,z)\ dS = \int \int_{D} f \{x,y,g(x,y)\}\ \sqrt{1 + (\frac {\partial g}{\partial x})^{2} + (\frac {\partial g}{\partial y})^{2}}\ dx\ dy[/itex]

## The Attempt at a Solution

This is how I have set it up so far. I seem to have gotten stuck, and it makes me think that there is something wrong with my set-up.

[itex]dS=\sqrt{4x^2+4y^2+1}dA[/itex]

[itex]\int\int(x^2+y^2)*2*\sqrt{x^2+y^2+\frac{1}{4}}dA[/itex]

[itex]2\int_0^{2\pi}\int_0^2(r^2)*(r^2+\frac{1}{4})^{1/2}rdrd\theta[/itex]

[itex]2\int_0^{2\pi}\int_0^2(r^3)*(r^2+\frac{1}{4})^{1/2}drd\theta[/itex]

Please let me know if you see any errors in my work, and if I am on the right track perhaps a gentle nudge on how to integrate this mess.

Thanks,

Mac

Last edited: