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→The Surface Integral
=The Surface Integral=
=The Surface Integral=
==Definition==
Now we turn to the idea of integrating a three-variable function over a surface, ''S''. This is called a surface integral. As we always do, we divide this region into ''n'' pieces <math>S_k, 1\leq k \leq n</math>, and we pick a point <math>(x_k,y_k,z_k)\in S_k</math> for each piece. Letting the area of <math>S_k</math> be <math>A_k</math>. The definition follows as expected:
<math>\iint_S f(x,y,z)dL = \lim_{A_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k,z_k)A_k </math>
==Computation==
Let <math>x=x(u,v),y=y(u,v),z=z(u,v) </math> be a parametric description of <math>S </math>, so that ''S'' will be defined as the image under these functions these three functions ''x,y,z'' of some region R of the ''uv'' plane.
As before, the trick to finding a convenient formula for a surface integral is finding a formula for that surface area element, ''dA''.
{{MVCalc}}
{{MVCalc}}