I have been given this question to solve however I'm having some difficulty solving it as I am quite new to Partial Differential Equations:

Let $ a = (2,2) $ and $ r =5 $

Compute the following surface integral

$$ A := \int_{\partial{B}(a,r)}\frac{x}{x^2+y^2}{dS}$$

I have been given the hint: try to use the mean value formula rather than computing the actual surface integral.

I have the mean value formula in my notes as:

$$ u(x) = {\int\!\!\!\!\!\!-}_{B(x,r)}u(y)dy = {\int\!\!\!\!\!\!-}_{\partial B(x,r)}u(y)dS(y)$$

However I'm struggling to apply this formula to the given function in the question

  • 1
    $\begingroup$ It seems to me that there are too many x's in your notes - it should be u(y) dS(y) in the integrand. Hence A is u(a) times some volume element (check the definition of the integral!). $\endgroup$ – Stockfish Oct 15 '18 at 19:29

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