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

Therefore
E = { (x, y, z) I -2 ~ X~ 2, x 2 ~ y ~ 4, o' ~ z ~ 2- h}
= { (x,y,z) I 0 ~ y ~ 4, -Jv ~ x ~ V"Y. 0 ~ z ~ 2 - tv}
= { (x,y,z)I O ~y~4,0 ~z~ 2-ty , -vv~ x~ JY}
= { (x, y, z) I 0 ~ z ~ 2, 0 ~ y ~ 4- 2z, - VY ~ x ~ vY}
SECTION 15.7 TRIPLE INTEGRALS 0 273
= { (x,y,z) l-2 ~ x ~ 2, 0 ~ z ~ 2 - tx 2 , x 2 ~ y ~ 4 - 2z}
= { (x, y, z) I 0 ~ z ~ 2, - v'4 - 2z ~ x ~ y'4- 2z, x 2 ~ y ~ 4 - 2z }
Then 1JJ~ f(x, y, z ) dV = I~ 2 I:2 I~ - 1112 f(x,y,z) dzdydx = I 0
4
I~ I 0
2
-v
12
f (x, y, z) dz dxdy
33. y
= 1~ 4 J~ - v/ 2 1~f(x , y ,z) dxdz dy =I~ Io 4 - 2 " J~f (x,y,z)dxdydz
2 f 2 - %2 / 2 !4- 2::
J
J( ) d d d r2 Jv'4=2z f4- 2z ( ) .
= -2 Jo z2 x , y,z y z ·X= Jo - "'4 - 2z z2 I x ,y ,z dydxdz
The diagrams show the projections
of Eon the xy-, yz-, and xz-planes.
-~------+--+
0 I X
Therefore
Io 1 1
I{z 1~l - v f (x, y, z ) dz dydx =I: I/ I:-rJ f(x, y, z) dzdx dy = f 0
Iol-,z It f(x, y, z) dx dy dz .
rt rt- u1'tJ 2 J( ) d d rt rt-../XJ1- z (
= JoJo 0 x,y,z x zdy= Jo Jo ..;x f x,y,z)dydzdx
fl /·(l - z) 2 Jl-z ( )
= J o. 0 ..;x f x,y,z dydxdz
35.
y
z
X
I: I:Icf f(x,y, ~)dzdxdy = IIIE f(x, y ,z)dV where E = {(x,y,z) I 0 ~ z ~ y, y ~ x ~ 1, 0 ~ y ~ 1}.
If D 1, D2, and D3 are the projections of Eon the xy-, yz- and xz-planes then
Dt = [(x, y) I 0 ~ y ~ 1,, y ~ x ~ 1} = {(x, y) I 0 ~ x ~ 1, 0 ~ y ~ x},
D2 = [(y,z) I 0 ~ y ~ 1, 0 ~ z ~ y} = {(y,z) I 0 ~ z ~ 1, z ~ y ~ 1}, an~
D3 = [(x,z) I 0 ~ x ~ 1, 0 ~ z ~ x} = { (x, z ) I 0 ~ z ~ 1, z ~ x ~ 1}.
[continued]
® 2012 C ngage Lc.-.uning. AU Rights RcscrYed. Mtty DOl be scanned. copied. or duplicated. or posted too publicly accessible website:, in whole or in part.