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

884 Chapter 27 Line integrals and multiple integrals
If we wish to change the order in which the integration is carried out, care must be
takenwiththelimitsofanon-rectangularregion.ConsideragainFigure27.12.Theline
x = 2 − 2y can be written asy = 2 −x . We can describe the regionRby restricting
2
attention to the vertical strip 0 x 2, and then letting y vary from 0 up to 2 −x
2 ,
that is
∫ x=2 ∫ y=(2−x)/2
x=0
y=0
4x+5dydx
You should check by evaluating this integral that the same result is obtained as in
Example 27.12.
Example27.13 Evaluate the double integral of f (x,y) = x 2 + 3xy over the regionRindicated in Figure
27.13by
(a) integrating first with respecttox, and thenwith respecttoy,
(b) integrating first with respecttoy, and thenwith respecttox.
Solution (a) If we integrate first with respect toxwe must select an arbitrary horizontal strip as
shown in Figure 27.13 and integrate in the x direction. On OB, y = x so that the
lower limit of thexintegration isx =y. On AB,x = 1 so the upper limit isx = 1.
Therefore
∫ x=1
[ ] x
x 2 3
+3xydx=
x=y 3 + 3x2 y 1
2
y
( 1
=
3 + 3y ) ( ) y
3
−
2 3 + 3y3
2
= 1 3 + 3y
2 − 11y3
6
Asyvaries from 0 to 1 the horizontal strips will cover the entire region. Hence the
limitsof integration of theyintegral are 0 and 1.So
∫ y=1
y=0
1
3 + 3y
2 − 11 6 y3 dy=
[ 1
3 y + 3 ] 1
4 y2 − 11y4
24
0
= 1 3 + 3 4 − 11
24
= 15
24
= 5 8
thatis,
∫ y=1 ∫ x=1
y=0
x=y
x 2 +3xydxdy= 5 8