James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1124 Chapter 16 Vector Calculus
that is,
rs, d − sin cos i 1 sin sin j 1 cos k
As in Example 16.6.10 , we can compute that
Therefore, by Formula 2,
| r 3 r | − sin
Here we use the identities
cos 2 − 1 2 s1 1 cos 2d
sin 2 − 1 2 cos 2
Instead, we could use Formulas 64 and
67 in the Table of Integrals.
y
S
y x 2 dS − yy ssin cos d 2 | r 3 r | dA
D
− y 2
y
sin 2 cos 2 sin d d − y 2
cos 2 d y
sin 3 d
0 0
0
0
− y 2
0
1
2 s1 1 cos 2d d y
− 1 2 f 1 1 2 sin 2g 0
2
0 ssin 2 sin cos2 d d
f2cos 1 1 3 cos3 g 0
−
4
3
■
Surface integrals have applications similar to those for the integrals we have previously
considered. For example, if a thin sheet (say, of aluminum foil) has the shape of a
surface S and the density (mass per unit area) at the point sx, y, zd is sx, y, zd, then the
total mass of the sheet is
and the center of mass is sx, y, zd, where
m − yy sx, y, zd dS
S
x − 1 y y x sx, y, zd dS
m
S
y − 1 y y y sx, y, zd dS z − 1 y z sx, y, zd dS
m m
S
Sy
Moments of inertia can also be defined as before (see Exercise 41).
Graphs of Functions
Any surface S with equation z − tsx, yd can be regarded as a parametric surface with
parametric equations
x − x y − y z − tsx, yd
and so we have
Thus
r x − i 1S
−xD −t k
r y − j 1S
−yD −t k
and
3 r x 3 r y − 2 −t
−x i 2 −t
−y j 1 k
| r x 3 r y | ÎS
−xD
− −z 2
1S
−yD
−z 2
1 1
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