Thomas Calculus 13th [Solutions]

1104 Chapter 15 Multiple Integrals
24. The area of the region R is 4 times the shaded in the following figure.
y
2
1
0 1 2
x
The area integral will be easy to compute in polar coordinates, but in rectangular coordinates the calculation is
awkward.
R
1
4 x
2
4
2 2
x
3 2 3 3
dA 4
2
1 dy dx 1dy dx 4 4 3
2 12 3 2 4
0
1 x
1
0
(As in Exercise 23, use integration by parts to evaluate the integrals
2
4 x dx and
2
1 x dx . )
Geometrically the area is the difference between the area of a circle of radius 2 and the area of a circle of radius
1, or 4 3 .
25.
5 0 y
10,000e 2 5 2 0 5
dy dx 10,000 1 e
dx
10,000 1 e
dx dx
5 2 1 5 1
5 1 0 1
x
x
2 2
2
0
2
5
x
e
x
2 5 2 0
10,000 1 e 2 ln 1 10,000 1 2 ln 1
x
x
2 2
2 5 2 5 2 7
2 2 2
10,000 1 e 2 ln 1 10,000 1 e 2 ln 1 40,000 1 e ln 43,329
26.
2 2
1 2 y y
1 2 y y 1 2 1 3
2 100( 1) 100( 1)
2
100( 1) 2 2 200
0 y
0 y
0 0
y dx dy y x dy y y y dy y y dy
2
y
4
y
1 1
2 4
0
4
200 (200) 50
27. Let ( x , y ) be the location of the weather station in county i for i 1, , 254. The average temperature in
i
i
254
T ( x , y ) A
i i i
i 1
Texas at time t 0 is approximately
, where T( x , )
A
i y i is the temperature at time t 0 at the weather
station in county i,
A is the area of country i, and A is the area of Texas.
i
28. Let y f ( x ) be a nonnegative, continuous function on [ a, b ], then
b
f ( x ) dx a
R
b f ( x) b f ( x)
a 0
a 0
A dA dy dx y dx
29. Since f is continuous on R , if m f ( x, y) M , property 3(b) of double integrals gives us
m dA f ( x, y)
dA M dA and hence mA( R) f ( x, y) dA MA( R).
R R R
R
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