I have attempted to convert to cylindrical coordinates and have gotten to the equation r*sqrt(4r^2+1) but I am unsure of how to set up the triple integral limits from here. If this is correct so far, how do you find the limits for each integral, if not then how would you go about solving it? Thanks.
Also, I know the answer is pi/6(37^(3/2)*-1) which is about 117.319
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$\begingroup$Your $x$ and $y$ should move inside the disk of radius 3 (from $x^2+y^2=3^2$). So \begin{align} \iint_D\sqrt{1+f_x^2+f_y^2}\,dA &=\int_0^3\int_0^{2\pi} \sqrt{1+4r^2}\,r\,d\theta\,dr\\ \ \\ &=2\pi\,\int_0^3 r\sqrt{1+4r^2}\,dr\\ \ \\ &\ \ \ \ \textit{(set $u=1+4r^2$)} \\ \ \\ &=\frac\pi4\,\left.\vphantom{\int}\frac23\,u^{3/2}\right|_1^{37} =\frac\pi6\,(37^{3/2}-1) \end{align}
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