Calculo 2 De dos variables_9na Edición - Ron Larson

1000 1000 Chapter CAPÍTULO Chapter 14 14 Multiple Multiple Integración Integration Integration múltiple
1000 1000 Chapter Chapter 14 14 Multiple Multiple Integration
z z
EJEMPLO EXAMPLE EXAMPLE 6 Encontrar 6 6Finding Finding el the valor the Average Average promedio Value Value of de a of una Function a Function función
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EXAMPLE 66 Finding the the Average 1 1Value of of a a Function
6
6
Encontrar Find Find the el average the valor average promedio value value of de f of x, fy
x, y 2 xy over
2 xy over sobre the region the la region región R, where R,
where donde R is R
a is rectangle es a un rectangle rectángulo
con vertices
with with
1
6
Find vértices vertices 0, 0
the average (0, , 0), 4, , 0
value (4, , 0), 4,
of , 3 (4, , and
f x, 3) y,
y and 0, 1
6
Find the average value of f x, y (0, 33).
2 xy 0, . 3 over .
2 xy over the the region region R, R, where where RRis is a rectangle a with with
5 5
vertices Solution vertices Solution
0, 0, 0 The 0 , , 4,
The area 4, 0 0 ,
area of , 4, 4, 3 the of 3 , and , the rectangular and rectangular
0, 0, 33 . . region region R is RAis A12
(see 12 (see Figure Figure 14.23). 14.23). The The
5
Solución El área de la región rectangular R es A 12 (ver la figura 14.23). El valor
5
Solution average average value The The value is area given
area is of given of the by
the rectangular by
region region RRis is AA 12 12(see (see Figure Figure 14.23). 14.23). The
4
The
4
promedio está dado por
1
average average value value is is given given by by 4 3
1
4 3
4
f(x, y) = xy
1 1
1 1 1
4
f(x, y) = xy
2
1
2
fx, fx, y dAy dA
1
4 3
f(x, y) = xy
1 A R 1 12 01
fx, y dA
A R
12 0 0 2 xy 0 2 xy dy dx 3
A R
12 0 0 2 xy dy dx 3
1
4 3
f(x, y) = xy
1
1 1
2 2
f y dA
4
A dy dx 4 3
3
R
121
0 10
12 xy dy dx 3
3
1
2 2
12
3
1 121
2
12 0 4 xy2 0 4 xy2 dx
0 4 xy2 dx
4 3
1 1 0 0
2
12
4 dx
1
4
0 4 xy2 dx
19
0
1
9 0
1
x dx
12 4
1 124
4 x dx
1 9 9 04
1 1 1
0
1
y y
x dx
4 x
4dx
(0, 0) (0, 0)
(0, 3) (0, 3)
31213
4 41
0
1
y
1
16 4
(0, 0)
(0, 3)
3 16 2 x2 0
1
y
1
4
1 20
x2 (0, 0)
(0, 0
2
R 3)
3 1
2
R
1 1
3
16 163
2 x2 2 x2 3
3 0
2
R
3
3 16 8
4
(4, 3)
16 8
0
2
R
4
(4, 3)
3 (4, 0) (4, 0)
3
x
16 8
4
x
(4, 3)
16 3 3
8
4
(4, 3)
(4, (4, 0) 0)
Figure 14.23 x
2 . Figura
Figure 14.23
14.23
3
Figure 14.23
2 . 2 . x
3
Figure 14.23
2 .
14.2 14.2Exercises See www.CalcChat.com See www.CalcChat.com for worked-out for worked-out solutions solutions to odd-numbered to odd-numbered exercises. exercises.
14.2 Exercises See See www.CalcChat.com for for worked-out worked-out solutions solutions to odd-numbered to odd-numbered exercises. exercises.
Approximation Approximation In Exercises In Exercises 1– 4, 1– approximate 4, approximate the the integral integral 6. Approximation 6. Approximation The The figure figure shows shows the level the level curves curves for a for function
f over f over a square a The square region
14.2 Ejercicios
a func-
Approximation
R fx, R fx, y dAy by dA In
dividing
In
by Exercises Exercises
dividing the
1– the rectangle
1– 4, 4,
rectangle approximate
R with R with vertices
the the
vertices integral
0, 0
integral
0, , 0 , 6. 6. Approximation The figure figure
region shows
R. Approximate
shows
R. the Approximate
the level level curves
the integral
curves
the for for
integral a function
f over a region R. Approximate the integral using
using
a function
using
4, 0
R fx,
4, , 4, 0
y
, 24, , and
dA
2
by
, and 0, 2
dividing
0, into 2 into eight
the
eight equal
rectangle
equal squares
R
squares and
with vertices
and finding finding the
0, 0 ,
the four four squares,
f over
squares, selecting
a square
selecting the midpoint
region
the
R.
midpoint of each of each square
the
square as x
integral
as i , y
using i x i
., y i .
R fx, y dA by dividing the rectangle R with vertices 0, 0 ,
8
Aproximación 8 En los ejercicios 1 a 4, aproximar la integral 6. Aproximación 2 2 2 2 La figura muestra las y
4, curvas y de nivel de una
sum 4, 00 , , 4, 4, 22 and 0, 2 into eight equal squares and finding the four squares, selecting the midpoint of each square as x
dividiendo el rectángulo R con vértices (0, 0), (4, 0), función ƒ en una región cuadrada R. Aproximar la integral i , y i .
R fsum fx , and
x, y dA i , fx y 0,
i i , yA 2
i
where into eight
A i where x i , yequal x i , is y i
the squares
is center and
the center of the finding
of ithe square. the four squares, selecting the midpoint of each square as x
i
i
ith square.
i , y i .
8
fx, empleando
0 0
8
fx, y dy y dx
i 1
dy dx
i 1
2
2
2
y
2
y
sum (4, Evaluate sum 2) y fx 0 0
Evaluate fx (0, the i , y i , i 2) iterated y i
en A
the iterated i Awhere ocho i
integral cuadrados x
integral i , xy i , and i yis i is the
and compare iguales the center
compare center it y of with hallando of the the with the i
ith square. approximation.
the la square. suma approxi-
x i , y i the A 0 0
2
fx, cuatro cuadrados y tomando el punto medio de cada
fx, y y dy dy dx
2
dx
i 1
cuadrado
0 0
como x
donde es el centro del cuadrado i-ésimo.
i , y i .
8 i 1
2
Evaluate fmation.
Evaluate the iterated iterated i
integral integral x i , y i and and compare compare it with with the the approximation.
2
approximation.
1
2 2
i1 4 2 4 2
4 2
4
2 2
y
4
Evaluar
4 2
2
1. 1. la integral iterada y compararla
1
2. con la aproximación.
1 6
x y dy dx 2.
x f x, y dy dx
4 2
4 2
4
1 4
2
00
2
1.
1.
2.
2.
1 6
8
x y dy dx
1 x
2 x y dy dx
x y dy dx
0 0
2
0 0
8
0
y dy dx
10
0
2 1 6
x
2
2 2
x y dy dx
y dy y dx
4 2
4 2
dy dx
8
4
0 0 0 0
1 0 0 0 0
1. 4 2 4 2
2.
1 6
x y dy dx
4 2 x4
2
1 1
10
3. 3. x 2
x 4. 0
dy dx
2
x
4 2
4 2
4 20
0
1
10 4
3.
3. x 2
2 4 2 0 0 x 1 y 1 1 2
dy dx
4.
4.
1 dy dx
1 6
x
x
0 0
0 0 x 1 y 1 dy dx
1 2
0
2 y 2 2
2 y dy dx
0 0
dy y 2 dx dy dx 4.
0 0
8
dy dx
x
4 2
4 2
0 0
0 0 x 1 y 1 10 1 2
3. x
0
0
2 y 2 dy dx 4.
dy dx
2 y 2 x
0 0 dy dx
0 0 x 1 y 1 1 2
5. Approximation 5. Approximation The The table table shows shows values
0values xof a 1y of function a function 1f
over f over a a In Exercises In Exercises 7–12, 7–12, sketch sketch the region the region R and R and evaluate 8
evaluate the iterated the iterated
5. 5. 5. Approximation Aproximación square square region region R.
The La Divide
The
R. tabla table Divide the
table muestra shows
region
shows
the values region valores into
values of into 16 de of a una equal
function a
16
function función equal squares
f squares fƒover sobre and
a a
and integral
10
x
una square región select
region cuadrada Divide R. Dividir the region la región into en 16 16 equal cuadrados squares iguales
In Exercises
integral R fx, 7–12, R fx, y dA.
sketch
y dA.
select x i , y x the region R and evaluate the iterated
i , to y
R. i
be to the be point the point the ith the square ith square closest closest to the In Exercises 7–12, sketch the region R and evaluate the iterated
square region i R. Divide the region into 16 equal squares and and
to the
1 2
y origin.
select elegir origin. Compare x to como be el the punto point más in cercano the al square origen closest en el cuadrado
the
integral R
2 fx, 1 y dA.
2
x i , , Compare y i this this approximation approximation with
i ith
with that that obtained obtained by using integral 2 R 1 fx, y dA.
2
select x to
by
the
using
origin. i-ésimo. point i , y
the
Compare
point i to be the point in the ith square closest to the
Comparar the in ith the square
this
ith
esta square farthest aproximación farthest from
with
from the con origin.
7.
that
the la obtained
origin.
7. 1 12x 2x2y dy 2y dx dy dx8.
8. sin 2 sin x cos
obtenida by usando
2 x 2 cos y dy 2 y dx
2
dy dx
2
1
1
2
origin. Compare this approximation with that obtained by using
2
using
0 0 0 0
0 0 0 0
el 4
the punto 4
the point 4 4
point más in the lejano ith square al origen farthest en el from cuadrado the origin. i-ésimo.
En 7. 6 3
los ejercicios 6 13
2x7 a 12, 2y dibujar dy dx la 8. región R y sin evaluar 2 x cos 2 la y integral dy dx
0 0 fx, in
0 0 fy the
x, dy ith
y dxdy square
dx
farthest from the origin.
7. 1 2x 2y dy dx 8. sin 2 x cos 2 y dy dx
0
0
0
0
0
0
0
0
4 4
iterada
9. 9.
6
3
R
x
f x, xy y
dx
dA. y dy
4 4
dx dy
0 0 f x, y dy dx
0 y 2
0 1 2 3 4
9.
2 14
x y y dx dy
2
0 y 2
0 1 2 3 4
7. 4
1 2x 2y dy dx 8. sin 2 x cos 2 y dy dx
y
x
0
y
x 0 y 2
0 1 2 3 4
4 y
x y 6 3
0 0 fx, y dy dx
9. x y dx dy
0 y 2
10.
y
10. x 2 y 2 xdx 2 y 2 dy
4
dx dy
1 y 1
x 0 y 0
0 32 32 01 31 2
3128 1 3
2823 24
sen
2316
163 4
0 2
0
0
10.
0 0
10. a x
a x a
06
1 3
4 y
2
9.
x y dx dy 10. x
1
0
2 y 2 dx dy
11. 0
2 y 2 xy dx 2 a 2 dx dy
x 2 dy
1
010 32
1 31 32 32 31
3130 28
3027 31 28 23
2722 232816
0
2215
16 15 23 16
11. 2
11.
y y x xy dy y dx
a a
dy dx
a
2 a
a 2 x 2 x 2 a 2 x
1 2 1
y2
31 3130 30 2227 15 22 15
1 0 1 0 x y dy 1dx
1 1 y
2 31 1 y 2
2 28 30
2827 27
2724 22
2419 15
1912
11.
x y dy dx
12
a
11. a
y
a
12. a
a
a 22 xx 12.
2 e x 2 y 2
e 2 dx x y dy dx dy e x y edx x y dy
1
dx dy
2 1
0
1
1
1
1
y
2 y
3 28
3 23 28 28 27
2322 27 24
2219 27 24 19
1914 192412
1412
7 7
19 12
0 y 01
y x y dy dx
12. a e
1 0
1 1y
12.
2x
1
x y 0 0 0 0
12. e x y dx 2 dx dy dy e x e xy
dx y dx dy dy
3
0
3 23 2322 22 19 1419 7 14 7
0
y
y
1
1
0
0
0
0
4 23
4 16 22
1615 19
1512 14
127 7
0 0
4
e xy dx dy e xy dx dy
4 16 16 16 15 15 1215 7 7120
0 7 0
0
y1
00