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396 CHAPTER 12 Vector Integral Calculus

z
P : (x(u , v ), y(u , v ), z(u , v ) j
j
j
j
j
j
j
v
R j (u , v ) j
j
Δv
Σ j
y
u Σ

x
Δu
FIGURE 12.16 Grid rectangle R j maps to a patch of
FIGURE 12.15 Forming a grid over D. surface j .

v apart. These lines form rectangles R 1 ,··· , R n that cover D. Each R j corresponds to a patch
of surface j , as in Figure 12.16. Let (u j ,v j ) be a point in R j . This corresponds to a point
P j = (x(u j ,v j ), y(u j ,v j ), z(u j ,v j )) on j . Approximate the mass of j by the density at P j
times the area of j . The mass of the shell is approximately the sum of the approximate masses
of these patches of surface:
n

mass of the shell ≈ δ(P j ) area of j .
j=1
But the area of j can be approximated as the length of the normal at P j times the area of R j :
area of j ≈ N(P j ) u v.

Therefore,
n

mass of ≈ δ(P j )N(P j ) u v
j=1
and in the limit as u → 0 and v → 0 we obtain

mass of = δ(x, y, z)dσ.

The center of mass of the shell is (x, y, z), where
1 1
x = xδ(x, y, z)dσ, y = yδ(x, y, z)dσ,
m m
and
1
z = zδ(x, y, z)dσ,
m
in which m is the mass.
If the surface is given as z = S(x, y) for (x, y) in D, then the mass is

2 2

∂S ∂S
m = δ(x, y, z) 1 + + dy dx.
D ∂x ∂y

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October 14, 2010 14:53 THM/NEIL Page-396 27410_12_ch12_p367-424