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

886 Chapter 27 Line integrals and multiple integrals
where,asbefore,theinnerintegralisperformedfirst,integratingwithrespecttoz,with
x andybeing treated asconstants. So
∫ (
x=1 ∫ y=1
] ) 1
I =
[xz +yz+ z2
dy dx
x=0 y=0 2
0
∫ x=1
(∫ y=1
= x+y+ 1 )
2 dy dx
=
=
x=0
∫ x=1
x=0
∫ x=1
x=0
y=0
[xy + y2
[ ] x
2 1
=
2 +x 0
x + 1 2 + 1 2 dx
2 + 1 ] 1
2 y dx
0
= 3 2
Considerationofthelimitsofintegrationshowsthattheintegralisevaluatedoveraunit
cube.
27.6.3 Green’stheorem
There is an important relationship between line and double integrals expressed in
Green’s theorem inthe plane:
If the functions P(x,y) and Q(x,y) are finite and continuous in a region of the
x--y plane, R, and on its boundary, the closed curveC, provided the relevant partialderivatives
exist and arecontinuousinand onC, then
∮
∫ ∫ ( ∂Q
Pdx+Qdy=
C
R
∂x − ∂P
∂y
)
dxdy
where the direction of integration alongC is such that the region R is always to
the left.
Theconditionsgiveninthetheoremarepresentformathematicalcompleteness.Most
of the functions that engineers deal with satisfy these conditions, and so we will not
consider these further. The important thing to note is that this relationship states that a
line integral around a closed curve can be expressed in terms of a double integral over
the region,R, enclosed byC.
Example27.15 (a) Evaluate ∮ C xydx + x2 dy around the sides of the square with vertices D(0, 0),
E(1,0),F(1,1)and G(0,1).
(b) Convert the lineintegral toadouble integral and verify Green’s theorem.