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I am given the equation for the height of a fence, which is h(x,y,z) = 4 + ((sinx)/2) + (y/3) +z. The line integral of this function will give the lateral surface area of one side of the fence. However, I need to find the total surface area of the fence; does that mean that I will have to double the line integral, or do I need to do more to take into account the top of the fence too?

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    $\begingroup$ The problem as stated is not clear. Ask your teacher. I would simply answer: "area of one side = xx, area of two sides = 2 xx$ and compute xx. The problem certainly cannot involve the top of the fence because that information isn't given. $\endgroup$ – David G. Stork Apr 11 at 0:26
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Your problem is much bigger than the one/two faces. The line integral of a function $$h:\Bbb R^3\longrightarrow\Bbb R$$ without more information doesn't makes any sense. And the same $h$ as the height of the curve doesn't makes any sense.

More information is required. Namely, the projection on the plane $XY$, say $$\gamma:[a,b]\longrightarrow\Bbb R^2.$$ Now, the (scalar) line integral of $$h:\Bbb R^2\longrightarrow\Bbb R.$$ makes sense and will be...

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