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I haven't done a surface integral in a while so I am asking to get this checked.

\(\mathbf{F} = \langle x, y, z\rangle\) and the surface is \(z = xy + 1\) where \(0\leq x\leq 1\) and \(0\leq y\leq 1\).

\(\hat{\mathbf{n}} = \nabla f/ \lvert\nabla f\rvert = \frac{1}{\sqrt{3}}\langle 1, 1, 1\rangle\)

\(dS = \frac{\lvert\nabla f\rvert dxdy}{\frac{\partial f}{\partial z}} = \sqrt{3}dxdy\)

\(\mathbf{F}\cdot\hat{\mathbf{n}} = \frac{1}{\sqrt{3}}(x+y+z) = \frac{1}{\sqrt{3}}(x+y+xy + 1)\)

\[

\int_0^1\int_0^1(x + y + xy + 1)dxdy = \frac{9}{4}

\]

So is this the correct integral I should I obtain or is there a mistake some where?

\(\mathbf{F} = \langle x, y, z\rangle\) and the surface is \(z = xy + 1\) where \(0\leq x\leq 1\) and \(0\leq y\leq 1\).

\(\hat{\mathbf{n}} = \nabla f/ \lvert\nabla f\rvert = \frac{1}{\sqrt{3}}\langle 1, 1, 1\rangle\)

\(dS = \frac{\lvert\nabla f\rvert dxdy}{\frac{\partial f}{\partial z}} = \sqrt{3}dxdy\)

\(\mathbf{F}\cdot\hat{\mathbf{n}} = \frac{1}{\sqrt{3}}(x+y+z) = \frac{1}{\sqrt{3}}(x+y+xy + 1)\)

\[

\int_0^1\int_0^1(x + y + xy + 1)dxdy = \frac{9}{4}

\]

So is this the correct integral I should I obtain or is there a mistake some where?

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