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

(b) The wedge in question is the shaded area rotated from B = 81 to B = 82.
Letting
V.i = volume of the region bounded by the sphere of radius P;
and the cone with angle ¢; (B = 81 to 82)
SECTION 15.1 0 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 0 285
and letting V be the volume of the wedge, we have
y
V = (1122- 1l21)- (V12 - Vn)
· = 1(82 - B1)[pg(1- cos¢ 2 ) - p~(l - cos¢ 1 )- p~(1- cos¢ 2 ) + p~(1- cos¢ 1
) ]
= 1(82- 81) [(P~- p~)(1- cos ¢ 2 ) - (P~- pi) (1- cos ¢1)] = i(B2 - (Jt)[(p~- pi) (cos ¢ 1 - cos¢ 2 )]
Or: Show that V =
1
021 P2 s in 21 r cot ¢ 1
01 p 1 s in 4> 1 r cot 4> 2
r dz dr dB.
(c) By the Mean Value Theorem with f(p) = p 3 there exists some p with p 1 ~ p ~ p 2 such that
f(p 2 ) - f(p 1 ) = !'(p)(p 2 - p 1 ) or p~ - p~ = 3Ji D.p. Similarly there exists¢ with ¢1 ~ ~ ~ ¢ 2
such that cos ¢ 2 - cos ¢ 1 ·= (-sin¢) D.¢. Substituting into the result from (b) gives
D. V = (p 2 D.p)(B2- B1)(sin¢) D.¢ = p 2 sin¢> D.pD.¢ D.B.
15.10 Change of Variables in Multiple Integrals
1. x = 5u- v, y = u + 3v.
TheJacobianis
a(x
~( ,y)
1)
=
l8xf8u 8xf8v I
= =5(3) -(-1)(1) =16.
v u,v 8yf8u oyfav 1 3
1
3. x = e-r sinB, y :;= er cos B.
5 -ll
8(x,y) 18xf8r 8xj8fJ I ~ -e - ''sinB e-rcosBI · ·
-- - - r . = e-•·er sin 2 B- e-re'· cos 2 B = sin 2 fJ- cos 2 ()or- cos 2()
8(r, B) - ayj8r ayjaB - er cos() -e smfJ . .
5. x = ufv, y = vfw, z = wfu.
a(x,y,z) -
a(u,v,w)-
oxfou 8xfav 8x/8w 1/ v - u/v 2 0
ayfau ayf8v 8yf8w 0 1/w -vjw 2
azjou 8z/8v 8z/8w - wfu2 0 1/u
= .!_11/w - vjw 2 ! ( u) I 0 -vjw 2 ! I 0 1/w I
v 0 1/u - - V 2 .:..wfu 2 1/ u + 0 -wfu 2 0
=.!. (...!_ - o) + ~ (o- ~) + o = - 1 - - - 1 - = o
v uw v 2 u 2 w uvw uvw
1. The transformation maps the boundary of S to the boundary of the imageR, so we first look at side 8 1 in the uv-plane. S 1 is
described by v = 0, 0.::; u.::; 3, sox = 2u + 3v = 2u andy = u- v = u. Eliminating u, we have.x = 2y, 0 _::; x ~ 6. 8 2 is
@ 20 12 Ccng:~ge Learning. All Right5 Resen-cd. MBy not be sccumed. copied. or duplicated. or posted 10 a publicly acccssiblo wcb~ itc, in whole or in prut.