Answer:
Step-by-step explanation:
Consider a surface f(x, y); you might temporarily think of this as representing physical
topography—a hilly landscape, perhaps. What is the average height of the surface (or
average altitude of the landscape) over some region?
As with most such problems, we start by thinking about how we might approximate
the answer. Suppose the region is a rectangle, [a, b] × [c, d]. We can divide the rectangle
into a grid, m subdivisions in one direction and n in the other, as indicated in figure 15.1.1.
We pick x values x0, x1,. . . , xm−1 in each subdivision in the x direction, and similarly in
the y direction. 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 should
be (depending on the fineness of the grid) close to the average height of the surface:
f(x0, y0) + f(x1, y0) + · · · + f(x0, y1) + f(x1, y1) + · · · + f(xm−1, yn−1)
mn
.
As both m and n go to infinity, we expect this approximation to converge to a fixed
value, the actual average height of the surface. For reasonably nice functions this does
indeed happen.
