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Evauate Surface integral

  • Thread starter boneill3
  • Start date
  • #1
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Homework Statement



Evauate Surface integral
[itex]\int\int_{\sigma}(x^2 + y^2)dS[/itex]
where [itex]\sigma [/itex] is the portion of the sphere [itex]x^2 + y^2 + z^2 = 4 [/itex]
above the plane z = 1.

Homework Equations


[itex]
\int\int_{\sigma} f(x,y) \sqrt{\frac{\partial z}{\partial x}^2+\frac{\partial z}{\partial y}^2+1}
[/itex]

The Attempt at a Solution



The plane and the sphere intercect at z=1 and x2+y2=3.
so
[itex]z=\sqrt{4-x^2-y^2}[/itex]

we have

[itex]
\frac{\partial z}{\partial x}= \frac{-x}{\sqrt{4-x^2-y^2}} [/itex]
and
[itex]\frac{\partial z}{\partial y}= \frac{-y}{\sqrt{4-x^2-y^2}} [/itex]

Z gives a projection of a disk onto the xy plane of x2+y2[itex]\leq 3[/itex]

Which is
[itex] R: 0 \leq \theta \leq 2\pi [/itex] [itex] 0\leq r \leq \sqrt{3} [/itex] in polar co-ordinates

Therefore
[itex]\int\int_{\sigma}(x^2 + y^2)dS[/itex]= [itex]\int\int_{R}(x^2 + y^2)\sqrt{(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2+1} [/itex]

which equals


[itex]\int\int_{R}(x^2 + y^2)dS[/itex]= [itex]\int\int_{R}(x^2 + y^2)\sqrt{(\frac{-x}{\sqrt{4-x^2-y^2}})^2+(\frac{-y}{\sqrt{4-x^2-y^2}})^2+1} [/itex]

In polar co-ordinates
We have I think.....

[itex]\int_{0}^{2\pi}\int_{0}^{\sqrt{3}}\left[(r^2)2\sqrt{\frac{-1}{r^2-4}} (r) \right]dr d\theta[/itex]

[itex]\int_{0}^{2\pi}\frac{-10}{3}d\theta[/itex]
[itex] = \frac{-20\pi}{3}[/itex]

Does this look alright ?
I'm not sure about if I've changed the [itex]\sqrt{(\frac{-x}{\sqrt{4-x^2-y^2}})^2+(\frac{-y}{\sqrt{4-x^2-y^2}})^2+1}[/itex] to polar co-ordinates correctly

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
That actually looks pretty good. Except where did you get that wacky (-1) factor? You are integrating a nonnegative function, x^2+y^2. The answer had better be nonnegative.
 
  • #3
127
0
Thanks alot!
I don't think I was concentrating enough.
It is very easy to get lost doing some of these integrals.
 
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