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

996 Chapter 15 Multiple Integrals
Example 7 Find the volume of the solid S that is bounded by the elliptic paraboloid
x 2 1 2y 2 1 z − 16, the planes x − 2 and y − 2, and the three coordinate planes.
16
12
z 8
4
0
0
1
y
2 2
1
x
0
SOLUTIon We first observe that S is the solid that lies under the surface
z − 16 2 x 2 2 2y 2 and above the square R − f0, 2g 3 f0, 2g. (See Figure 15.) This
solid was considered in Example 1, but we are now in a position to evaluate the double
integral using Fubini’s Theorem. Therefore
V − yy s16 2 x 2 2 2y 2 d dA − y 2
0 y2 s16 2 x 2 2 2y 2 d dx dy
0
R
− y 2
0 f16x 2 1 3 x 3 2 2y 2 xg x−0
x−2
dy
FIGURE 15
− y 2
0 (88
3 2 4y 2 ) dy − f 88
3 y 2 4 3 y3 g 0
2
− 48 ■
In the special case where f sx, yd can be factored as the product of a function of x only
and a function of y only, the double integral of f can be written in a particularly simple
form. To be specific, suppose that f sx, yd − tsxdhsyd and R − fa, bg 3 fc, dg. Then
Fubini’s Theorem gives
yy f sx, yd dA − y d
c
yb tsxdhsyd dx dy − y d
Fy b
tsxdhsyd dxG dy
a
c a
R
In the inner integral, y is a constant, so hsyd is a constant and we can write
y d
c
Fy b
tsxdhsyd dxG dy − y d
FhsydSy b
tsxd dxDG dy − y b
tsxd dx y d
hsyd dy
a
c
a
a
c
since y b tsxd dx is a constant. Therefore, in this case the double integral of f can be written
as the product of two single
a
integrals:
11 yy tsxd hsyd dA − y b
tsxd dx y d
hsyd dy where R − fa, bg 3 fc, dg
a c
R
Example 8 If R − f0, y2g 3 f0, y2g, then, by Equation 11,
yy sin x cos y dA − y y2
sin x dx y y2
cos y dy
0
0
R
− f2cos xg 0
y2
fsin yg 0
y2
− 1 ? 1 − 1 ■
The function f sx, yd − sin x cos y in
Example 8 is positive on R, so the
integral represents the volume of the
solid that lies above R and below the
graph of f shown in Figure 16.
z
0
y
FIGURE 16
x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.