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Unformatted text preview: Section 17.7 Surface Integrals “Integrating Functions over Arbitrary Surfaces” In the last few sections we were considering integrals of functions of two or three variables over arbitrary curves. In this section we consider the integral of a function of three variables over a surface. Note that usually we integrate functions of three variables over regions in 3-space. 1. Surface Integrals The definition of a surface integral is analogous to many of the other definitions of integrals we have given - it will require partitioning the surface into small rectangles and then constructing a Riemann sum over those rectangles. We proceed as follows. ( i ) Suppose f ( x, y, z ) is some function of three variables and S is a piecewise smooth surface in 3-space on which f is continuous. ( ii ) Divide S up into small patches of area Δ S . ( iii ) In each patch, fix a point ( x i , y j , z k ). ( iv ) Construct the Riemann sum n summationdisplay i =1 f ( x i , y j , z k )Δ S. ( v ) We define the Surface Integral of f ( x, y, z ) over S to be the limit of the sum integraldisplay integraldisplay S f ( x, y, z ) dS = lim n →∞ n summationdisplay i =1 f ( x i , y j , z k )Δ S provided this limit exists. Surfaces have been described in many different ways, and accordingly, we have many different ways to evaluate surface integrals depending upon description. We shall look at the most important ways. 1.1. Graphs of Functions. Suppose that the surface S is the graph of some function z = g ( x, y ) over the region D in the xy-plane. Then the surface integral of f ( x, y, z ) over S can be calculated as integraldisplay integraldisplay S f ( x, y, z ) dS = integraldisplay integraldisplay D f ( x, y, g ( x, y )) radicalBigg parenleftbigg ∂g ∂x parenrightbigg 2 + parenleftbigg ∂g ∂y parenrightbigg 2 + 1 dA....
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