Surface Integrals

We can consider these much like we did line integrals. Instead of integrating along a curve, we know integrate along a surface.

Let h(x,y,z) be a continuous vector field on a smooth surface
z=f(x,y), where (x,y) is in some plane region.

For line integral of g(x,y) over the curve r(t) we had
Integral of g(r(t)).dr, where dr is a portion of arc length.

For our surface integral, we have Integral over the region of h(x,y,f(x,y))dA
where dA is a portion of surface area, that is,
dA=Sqrt[1+(Dxf)^2+(Dyf)^2] dx dy

Note: some texts use dsigma or dS for dA and reserve dA for dydx,
a portion of PLANE area. We will use dA as above.

Example

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