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

272 0 CHAPTER 15 MULTIPLE INTEGRALS
Therefore
E = { (x y z) 1 -2 < x < 2 0 < y < 4 ·_ x 2 - 1 yl 4 - :z;2 - y < z < 1 yl 4 - :z;Z - y }
, , - -'-- , 2 - - 2
= { (x, y,z) I 0:::; y:::; 4, - y'4""='1j:::; x:::; y'4""='1j, - h/4 - x 2 - y:::; z ~ ~yf4- x 2 - y}
= { (x,y,z) I -1 $ z 5: 1, 0 5: y 5:4 - 4z 2 , - yf4 - y- 4z 2 :::; x:::; yf4 - y - 4z 2 }
= { (x, y, z) I 0 5: y 5: 4, - ty'4""='1j $ z $ ty'4""='1j, - j4-y- 4z 2 :::; x:::; yf4 - y- 4z 2 }
Then
= { ( x, y, z) I - 2 5: x 5: 2, - ~ J 4 - x 2 :::; z :::; t J 4 - x~, 0 :::; y :::; 4 - :z: 2 - 4z 2 }
= { (x, y, z) I -1 5: z 5: 1, - .J4 - 4z 2 :::; x:::; .J4- 4z 2 , 0:::; y:::; 4- x 2 - 4z 2 }
Iff sf ( x,y,z ) dV = !_ 2 ; -~/ 0 2 / x,y,z dzdydx=:Jo --14-ii -~ 1 2
2 r4-x2 JJ4-x2-yf2 ( ) r4 f-14-ii JJ4-:z:2-yj 2 ( )
! x,y,z dzdxdy
I I f4- 4z2 JV4-y-4z2 ( ) , f4 f-/4-!i/2 J V4-y-4z 2 (
= - tJo -.J _ _ z 4 11 4 2 f x, y, z dx dy dz = Jo --14-ii/ 2 - J _ _ : 4 11 4 2 f x, y, z) dx dz dy
f 2 J~/2 r4-x2 -4:2' . ) fl J J4- 4:;2 r4-o: 2 -4: 2 ( )
= _ 2
r;--:; Jo f(x,y,z dydzdx= _ 1 ~Jo f x,y,z dydxdz
-y4- :z:2/2
-y4-4z2
31 .
y
(- 2,4,0;
JC
(2,4,0)
X
4 y - 2 2 JC
If D1, Dz, and Da are the projections of Eon the xy-, yz-, and xz-planes, then
D1 = { (x, y) I -2 $ x 5: 2, :z: 2 ~ y 5: 4} = { (x, y) I 0 :::; y :::; 4, -y'Y $ x:::; vY },
D2 = { (y,z) I 0 :::; y $ ,4, 0 :::; z 5: 2- ~y } = { (y,'z) I 0:::; z 5: 2,0 ~ y:::; 4 -
2z}, and
Ds = { (x, z) I - 2 $ x :::; 2, 0 5: z ~ 2 - ~~ 2 } = { (x, z) I 0 :::; i ~ 2, - .J4- 2z :::;. x $ v'4 - 2z}
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