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

Section 15.2 Double Integrals over General Regions 1007
Property 9 can be used to evaluate double integrals over regions D that are neither
type I nor type II but can be expressed as a union of regions of type I or type II. Figure 18
illustrates this procedure. (See Exercises 57 and 58.)
y
y
D
D
D¡
0
x
0
x
FIGURE 18
(a) D is neither type I nor type II.
(b) D=D¡ D, D¡ is type I, D is type II.
The next property of integrals says that if we integrate the constant function f sx, yd − 1
over a region D, we get the area of D:
10 y 1 dA − AsDd
D
z
0
z=1
Figure 19 illustrates why Equation 10 is true: A solid cylinder whose base is D and
whose height is 1 has volume AsDd 1 − AsDd, but we know that we can also write its
volume as yy D
1 dA.
Finally, we can combine Properties 7, 8, and 10 to prove the following property. (See
Exercise 63.)
x
FIGURE 19
Cylinder with base D and height 1
D
y
11 If m < f sx, yd < M for all sx, yd in D, then
mAsDd < y f sx, yd dA < MAsDd
D
ExamplE 6 Use Property 11 to estimate the integral yy D
e sin x cos y dA, where D is the
disk with center the origin and radius 2.
SOLUtion Since 21 < sin x < 1 and 21 < cos y < 1, we have
21 < sin x cos y < 1 and therefore
e 21 < e sin x cos y < e 1 − e
Thus, using m − e 21 − 1ye, M − e, and AsDd − s2d 2 in Property 11, we obtain
4
e
< y e sin x cos y dA < 4e
D
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