Evaluate the following integral $$\int \int_S z^2 dS$$ where $S$ is the surface of the cube $[-1,1] \times [-1,1] \times [-1,1]$

My thoughts
I'm quite lost here. How do I know the projection and which vectors do i take to find the normal when there are no vectors? I really am in need of some urgent help please

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    $\begingroup$ Related math.stackexchange.com/questions/420820/…. $\endgroup$ – Ataraxia Jun 15 '13 at 17:16
  • $\begingroup$ @ZettaSuro Hi Zetta, thankyou for your response. I read the following post and I am still rather confused. We haven't learnt the divergence thereom yet, only surface integrals and we were given this question, I tend to understand more by seeing the example worked out, if its not too much of a hassle, is it possible if you could show me the working out? $\endgroup$ – amanda Jun 15 '13 at 17:26
  • $\begingroup$ Yea, I just thought it was worth mentioning. Actually, in that question we determined that the best way to do it is without the divergence theorem :) $\endgroup$ – Ataraxia Jun 15 '13 at 17:29
  • $\begingroup$ Any feedback is always helpful :) $\endgroup$ – amanda Jun 15 '13 at 17:38

Do it one surface at a time. To take the surface integral of a scalar field:


Where $z=g(x,y)$ defines the surface.

Start with the upper and lower faces ($z=\pm1$).




Since $f(x,y,z)$ is symmetric over the xy plane, $A_2$ will be the same.

Now do the second set of faces (the ones parallel to the xz-plane):




Finally, the last set of faces (the ones parallel to the yz-plane):




The total surface area is the sum $A_1+A_2+A_3+A_4+A_5+A_6$.

  • $\begingroup$ Thankyou so so much, you have made that so clear :)). $\endgroup$ – amanda Jun 17 '13 at 4:50

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