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

SECTION 15.7 TRIPLE INTEGRALS D 271
23. (a) The wedge can be described as the region
D = {(x,y,z) I y 2 + z 2 ::; 1, 0 ::; x::; 1, 0 ::; y ::; x}
= { (x, y , z) I 0 ::; x ::; 1, 0 ::; y ::; x, 0 ::; z ::; } 1 - .Y 2 }
So the integral expressing the volume of the wedge is
JffodV=fo1 fo"'J/1 -v2 dz dydx.
(b) C S . r l f"' n /l.-vz d d d " l
A A gives Jo Jo Jo z y x = 4 - 3·
(Or use Formulas 30 and 87 from the Table of Integrals.)
25. Here f(x, y, z) = cos(xyz) and~ V = ~ · t · ~ = ~,so the Midpoint Rule gives
1 [!(1 1 1) J(l 1 3) !(1 3 l) f(l 3 3)
= s 4 • 4•4 + 4•4• 4 + 4•4• 4 + 4 • 4 •4
+ !(~ , t. t) + !(~ , t. ~) + !(~, ~, t) + ! (~, ~. ~ ) ]
= ~ [cos -l4 +cos ~ +cos 6~ +cos f4: +cos 6~ +cos i 4 +cos f4: + cos~~ ) ~ 0.985
27. E = {(x, y, z) I 0 ::; x::; 1, 0::; z::; 1- x, 0 ::; y::; 2- 2z },
the solid bounde~ by the three coordinate planes and the planes
z = 1 - x, y = 2 - 2z.
X
29.
y
4
X
I
---~"--x' + 4z 1 = 4
.t
- I
If D 1 , D 2 , D 3 are th~ projections of Eon the xy-, yz-, and xz-planes, then
D 1 = {(x, y) I - 2::; X::; 2, 0 ::; y::; 4 - x 2 } = {(x, y) I 0 ::; y::; 4, - v4 - y::; X::; .J4=Y }
I .
D2 = {(y,z) 1 o ::; v ::; 4, - ~v4- y::; z::; ~v4- v} = {(y,z) 1-1 ::; z::; 1, o::; v ::; 4 - 4z 2 }
DJ = {(x,z) I x 2 +4z 2 ::; 4}
[continued]
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