# Evaluating $\iint_S z \, dS$

My professor gave us a handout for these problems and I'm not sure if I'm doing these correctly. Could someone tell me if I'm on the right track?

Problem

$\iint_S z \,\mathrm{d}S$

$z=x^2+y^2$ and $x^2+y^2\leq 4$

Attempt

$z=r^2$

$0\leq r\leq 2$

$x=\cos(\theta)$ and $y=\sin(\theta)$

$\iint_s f(x,y,z)\mathrm{d}S = \iint_D f(x,y,g(x,y))\sqrt{\frac{\mathrm{d}z}{\mathrm{d}x}^2+\frac{\mathrm{d}z}{\mathrm{d}y}^2+1} \,\mathrm{d}A$

$\int_0^{2\pi} \int_0^2 r^3 \sqrt{4\cos^2(\theta) + 4\sin^2(\theta)+1} \,\mathrm{d}r\,\mathrm{d}\theta$

$\sqrt5 \int_0^{2\pi} \int_0^2 r^3\,\mathrm{d}r\mathrm{d}\theta$

After integrating I get $8\pi\sqrt5$

• This is fine... – Faraad Armwood Dec 5 '16 at 4:38