(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]\int\int{\frac {x}{\sqrt {1+4\,{x}^{2}+4\,{y}^{2}}}}dS[/tex]

Where S is the parabaloid [tex] z = 25 - x^{2} -y^{2}[/tex] that lies within the cylinder [tex]x^{2}+(y-1)^{2}=1[/tex]

3. The attempt at a solution

First i use the following:

[tex]{\it dS}=\sqrt {1+{\frac {{{\it df}}^{2}}{{{\it dx}}^{2}}}+{\frac {{{

\it df}}^{2}{\it }}{{{\it dy}}^{2}}}}dA[/tex]

(**The above derivatives are not second derivatives, it should be each derivative squared)

to find

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

my integral then simplifies to the following, using polar coordinates and the following parametric equations

[tex]x = cos(\theta),and,

y = sin(\theta) + 1, and, dA = rdrd\theta[/tex]

[tex]\int\int cos(\theta) r dr d\theta[/tex]

This is where my trouble starts -- the integral is easy to evaluate, but I don't know how to set up my boundaries so that I am in fact integrating around a cylinder that has been shifted up in the xy plane.

The limits I would use (but which i do not think are right) are as follows:

[tex]2\pi\geq\theta\geq0[/tex]

[tex]1\geq r \geq 0[/tex]

once again, these limits do not take into account the translation of the cylinder on the xy plane. If anyone oculd please tell me where I went wrong or what I should do I would appreciate it greatly.

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# Surface Integral Help

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