Multiple Integration - Whitman College

15 Multiple a surfacef(x, y); you might temporarily think of this as representing physicaltopography a hilly landscape, perhaps. What is the averageheight of the surface (oraverage altitude of the landscape) over some region?As with most such problems, we start by thinking about how we might approximatethe answer. Suppose the region is a rectangle, [a, b] [c, d]. We can divide the rectangleinto a grid,msubdivisions in one direction andnin the other, as indicated in figure pickxvaluesx0,x1,..,xm 1in each subdivision in thexdirection, and similarly intheydirection. At each of the points (xi, yj) in one of the smaller rectangles in the grid,we compute the height of the surface:f(xi, yj). Now the average of these heights shouldbe (depending on the fineness of the grid) close to the averageheight of the surface:f(x0, y0) +f(x1, y0) + +f(x0, y1) +f(x1, y1) + +f(xm 1, yn 1) bothmandngo to infinity, we expect this approximation to converge to a fixedvalue, the actual average height of the surface.

388 Chapter 15 Multiple Integration Of course, for different values of yi this integral has different values; in other words, it is really a function applied to yi: G(y) = Zb a f(x,y)dx. If we substitute back into the sum we get nX−1 i=0 G(yi)∆y. This sum has a nice interpretation. The value G(yi) is the area of a cross section of the

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