Strategy for Surface Integrals

In a Nut Shell: The strategy used to evaluate surface integrals depends on whether you use

direct evaluation of the integral or you choose to use the method of transformation.

Note: If the surface, S, is described by z = g(x,y) then you will integrate in domain

D in the x-y plane.

On the other hand, if the surface is described by y = g(x,z), then D would be

the “area” projected on to the x-z plane and integration will be in the x-z plane. etc.

Strategy for Option 1: Use direct evaluation of the surface integral.

I_S = ∫ ∫ f(x,y,z) dS = ∫ ∫ F ∙ n dS = ∫ ∫ ∫ div (F) dV (from Divergence Theorem)

S S E

1.	Construct a view of surface, S, showing element of surface area, dS.
2.	Express dS in terms of parameters on the surface of S.
3.	Express f(x,y,z) in terms of parameters on the surface, S.
4.	Evaluate ∫ ∫ f(x,y,z) dS S

Click here for discussion of the strategy for Option 2 using a transformation process.

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