Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

288 0 CHAPTER 15 MULTIPLE INTEGRALS
2 2 2
(b) If we approximate the surface. of the earth by the ellipsoid 6
;., 82
+ 6
i 782
+ 6
; 562
_ = 1, then we can estimate
the volume of the earth by finding the volume of the solid E enclosed by the ellipsoid. From part (a), this is
fffE dV = ~11"(6378)(6378)(6356) ~ 1.083 X 10 12 km 3 •
(c) The moment ofintertia about the z-axis is I% = JJJE (x 2 + y 2 ) p(x, y, z) dV, where E is the solid enclosed by
xz yz zz 18(x y z) I
2 + b 2
+ 2 = 1. As in part (a), we use the transformation x = au, y = bv, z = cw, so 8
( ' ' ) = abc and
a . c u,v,w
I , = JJJE (x 2 + y 2 ) kdV = f.(.[ k(a 2 u 2 + b 2·u 2 )(abc) dudv dw
u2+ v2+w2 ~ 1
= abck f 0
" f 0
2 " J 0
1 ( a
2 p
2
sin 2 ¢ cos 2 () + b 2 p 2 sin 2 ¢ sin 2 8) p 2 sin ¢ dp d8 drjJ
= abck [a 2 f 0 ,.. f~,.. f 0
1 (p 2 sin
2
rP cos 2 B) p 2 sin 1/> dp d() dlj> + b 2 f 0 " f~,.. f 0
1 (p
2
sin 2 rjJ sin 2 B) p 2 sin rP dp dB dr!>]
=. a 3 bck J 0
" sin 3 rjJ drjJ f 0
2
" cos 2 B dB J; p 4 dp + ab 3 ck fo.,.. sin 3 rjJdrjJ f~" sin 2 B dB J; p 4 dp
= a 3 bck [ ~ cos 3 ¢-cos I/>] ~ [ ~B + i sin2 B]~" [iP 5 ] ~ + ab 3 ck [t cos 3 rjJ- cosr/J]~ [~B- i sin 28] ~" [tP 5 ]~
= a 3 bck (t) (1r) ( t) + ab 3 ck (1) (1r) (i) = rt7r (a 2 + b 2 ) abck
23. Letting u = x - 2y and v = 3x - y, we have x = f.(2v- u) and y = f(v- 3u). Then 8 8 ((x, y)) = ~ - 1 / 5 215 1 = .!.
0 0
u,v -3/5 1/5 5
and R is the image of the rectangle enclosed by the lines u = 0, u = 4, v = 1, and v = 8. Thus
8(xy) 1.:...1/2 1/ 21 1
25. Letting u = y- x, v = y + x, we nave y = !(u + v), x = Hv - u). Then (u: v) = i = 8 1 2 112
- and R is the
2
image of the trapezoidal region with vertices ( - 1, 1), ( -2, 2), {2, 2), and {1, 1). Thus
y -x
!h 121" u I 11 11 2 [ u]u=v 112
R Y +X '1 -v V 1 V u= -v 1
cos--dA-= cos - - - 2
du dv = - 2
v sin- dv = - 2
2v sin{1) dv = ~sin 1
.
27. Letu=x+yandv=-x+y.Thenu+v = 2y => v=Hu+v) andu - v=2x => X=~(u-v).
8(x y) 11/2 -1/ 2! 1
- 8
( ') = =- 2
.Now[u[= [x+y[< [x[+[y[ - l