082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

27.6 Double and triple integrals 885
y
y
1
B
1
B
x = y
y = x
y = x
R
dy
x = 1
R
O 1
A
x
0 y = 0 1
A
x
Figure27.13
Theintegral with respect toxhas
limitsx =yandx = 1.
Figure27.14
We integrate along a vertical strip
fromy=0toy=x.
(b) If we choose to integrate with respect toyfirst we must select an arbitrary vertical
stripasshowninFigure27.14.Atthelowerendofthestripy = 0.Attheupperend
y=x.
To integrate along the stripweevaluate
∫ y=x
y=0
] x
x 2 +3xydy=
[x 2 y + 3xy2 =x 3 + 3x3
2
0
2 = 5x3
2
Weaddcontributionsofallsuchverticalstripsbyintegratingwithrespecttoxfrom
x=0tox=1:
∫ 1
0
5x 3
[ 5x
4
2 dx = 8
] 1
0
= 5 8
We see that the prudent selection of the order of integration can yield substantial
savings inthe effort required.
27.6.2 Tripleintegrals
The techniques we have used for evaluating double integrals can be generalized naturally
to triple integrals. Whereas double integrals are evaluated over two-dimensional
regions, tripleintegrals areevaluated over volumes.
Example27.14 Evaluate
I =
∫ x=1 ∫ y=1 ∫ z=1
x=0 y=0 z=0
x+y+zdzdydx
Solution What ismeantby thisexpression is
I =
∫ x=1
(∫ y=1
(∫ z=1
x=0
y=0
z=0
) )
x+y+zdz dy dx