Calculus- Early Transcendentals, 2021a

Chapter 14
Multiple Integration
14.1 Volume and Average Height
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 14.1. We pick x values x 0 , x 1 ,..., x m−1 in each
subdivision in the x direction, and similarly in the y direction. At each of the points (x i ,y j ) in one of the
smaller rectangles in the grid, we compute the height of the surface: f (x i ,y j ). Now the average of these
heights should be (depending on the fineness of the grid) close to the average height of the surface:
f (x 0 ,y 0 )+ f (x 1 ,y 0 )+···+ f (x 0 ,y 1 )+ f (x 1 ,y 1 )+···+ f (x m−1 ,y n−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.
y 3
d
y 2
Δy
y 1
c
Δx
a x 1 x 2 x 3 x 4 x 5 b
Figure 14.1: A rectangular subdivision of [a,b] × [c,d].
Using sigma notation, we can rewrite the approximation:
1
mn
n−1
∑
i=0
m−1
∑ f (x j ,y i )=
j=0
=
1
(b − a)(d − c)
1
(b − a)(d − c)
n−1
∑
i=0
n−1
∑
i=0
m−1
∑
j=0
f (x j ,y i ) b − a d − c
m n
m−1
∑ f (x j ,y i )ΔxΔy.
j=0
The two parts of this product have useful meaning: (b − a)(d − c) is of course the area of the rectangle,
and the double sum adds up mn terms of the form f (x j ,y i )ΔxΔy, which is the height of the surface at a
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