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

SECTION 15.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 0 281
11. 2 :S p :S 4 represents the solid region between and including the spheres of
radii 2 and 4, centered at the origin.
0 :S ¢> :S f restricts the solid to that
portion on or above the cone ¢> = f, and 0 :S 8 :S 1r further restricts the
solid to that portion on or to the right of the xz-plane.
13. p :::; 1 represents the solid sphere of radius I centered at the origin.
3 ; :S ¢> :S 1r restricts the solid to that portion on or below the cone ¢> = ~;.
15. z 2': Jx2 + y 2 because the solid lies above the cone. Squaring both sides ofthis inequality gives z 2 2': x 2 + y 2 · :::?
2z 2 2:_ x 2 + y 2 + z 2 = p 2 :::? z 2 = p 2 cos 2 ¢> 2': ~p 2 :::? cos 2 ¢> 2': ~· The cone opens upward so that the inequality is
cos¢> 2': ~, or equivalently 0 :S ¢> :S *. In spherical coordinates the sphere z = x 2 +. y 2 + z 2 is p cos lp = p 2 :::?
p = cos¢. 0 :S p :S cos¢> because the solid lies below the sphere. The solid can therefore be described as the region in
spherical coordinates satisfYing 0 :S p :S cos ¢>, 0 :S ¢> :S *.
17.
The region of integration is given in spherical coordinates by
E = {(p, 8, ¢>) I 0 :S p :S 3, 0 :S 8 :S 7r / 2, 0 :S ¢> :S 1r / 6}. This represents the solid
region in the first octant bounded above by the sphere p = 3 and below by the cone
¢> = 1T/6.
J 0 " 16 J 0
" 12 .1; p 2 sin ¢> dp dB d4> = ./~" 16 sin 4> d¢> f 0 " 12 dB J: p 2 dp
X
= [-cos¢>] ~ 16 ( 8]~ 12 [~Pa]~
= (1- ~) G)(9) = 9 ; (2- v'3)
19. The solid E is most conveniently described if we use cylindrical coordinates:
E = {(r,B,z) I 0 :s e :s i.O :s r :s 3,0 :s z :s 2}. Then
JJJE f(x, y, z) dV = f 0 " 12 .1; ./~ f(r cos f)_, rsinB, z) 1' dz d1·d8.
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