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12.6 Applications of Surface Integrals 395

Then
2
√ 7 √

zdσ = 3 − x dx = 5 3.
2
0
SECTION 12.5 PROBLEMS

2
In each of Problems 1 through 10, evaluate 5. f (x, y, z)= z, is the part of the cone z = x + y 2

f (x, y, z)dσ. in the first octant and between the planes z =2and

z =4.
1. f (x, y, z) = x, is the part of the plane x + 4y + z = 6. f (x, y, z) = xyz, is the part of the plane z = x + y
10 in the first octant. with (x, y) in the square with vertices (0,0),(1,0),
(0,1) and (1,1).
2
2. f (x, y, z) = y , is the part of the plane z = x for
2
0 ≤ x ≤ 2, 0 ≤ y ≤ 4. 7. f (x, y, z) = y, is the part of the cylinder z = x for
0 ≤ x ≤ 2, 0 ≤ y ≤ 3.
3. f (x, y, z) = 1, is the part of the paraboloid
2
2
z = x + y 2 lying between the planes z = 2 8. f (x, y, z) = x , is the part of the paraboloid z =
2
2
and z = 7. 4 − x − y lying above the x, y - plane.
9. f (x, y, z) = z, is the part of the plane z = x − y for
4. f (x, y, z) = x + y, is the part of the plane
0 ≤ x ≤ 1and 0 ≤ y ≤ 5.
4x + 8y + 10z = 25 lying above the triangle
in the x, y - plane having vertices (0,0), (1,0) 10. f (x, y, z) = xyz, is the part of the cylinder z =
2
and (1,1). 1 + y for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
12.6 Applications of Surface Integrals
12.6.1 Surface Area
If is a piecewise smooth surface, then

dσ = N(u,v) du dv = area of .
D
This assumes a bounded surface having finite area. Clearly we do not need surface integrals to
compute areas of surfaces. However, we mention this result because it is in the same spirit as
other familiar mensuration formulas:

ds = length of C,
C

dA = area of D,
D

dV = volume of M.
M

12.6.2 Mass and Center of Mass of a Shell
Imagine a shell of negligible thickness in the shape of a piecewise smooth surface .Let
δ(x, y, z) be the density of the material of the shell at point (x, y, z). We want to compute the
mass of the shell.
Let have coordinate functions x(u,v), y(u,v), z(u,v) for (u,v) in D. Form a grid of
lines over D, as in Figure 12.15, by drawing vertical lines u units apart and horizontal lines

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October 14, 2010 14:53 THM/NEIL Page-395 27410_12_ch12_p367-424