CHAPTER 4
VECTOR
CALCULUS
INTEGRATION
,.
Consider same function 01 lxiy ) that exists in the Z plane
Where parametric equation
Ms
(x y) be
rlss
•
, can
expressed as a )
So we have 01 ( )
•
. \
x Z
y
lg
\ 1
.
New consider path C on the x -
y plane
which is parameterised by the variable It )
* We can new
integrate 01 an
the
path C
flx , y ) arz " " "
#I
*
Mt ) =
( xlt ) , ylt ) ) :
× (a)
←
,
.
yla )
f. #)
l.MY#lihikteEt:nmnnetYet
surface
x
t=b
Ib) ,
,
the
ylb )
beneath
area of
the
The
'
curtain
¢
'
b
fcads f HH ) dd÷
* =
0 at
a
, *
direction
*
a- b-
e.g. Integrate 01 =xiy2 from ) 11,0
to ( 0,1 )
along
the paths shown
'
.
*
for integration to work
(.cz
b- Co. 2) ° .
-
,
• C ,
the
it
,
is to
necessary express
, •
•
Cir parametric equations
'
, } as
c
y:
,
Q 2,0 )
Along as :
-
parametric farm
rlt ) =
a- + t( b- -
e) of a straight line
gip ;'
÷
"
pgaratmneterisf '
limits
:[6-
fcads
:
BHH )
fba ddttdt
=
b :c on )=a t.tt
ftp.tfgEY.jhmr#atuba
.
1
t
=
1
a :( 1,0 )
=
( l -
t ,t )
;':*
If
t O
III
Is
=
' "
antes
"
asayarattzn
=fz[t ¥+53 ]
1.
-
x= it
=f2[
2
( i tat
-
's ) -
( o )] Y2= A
rz ±
=
.
VECTOR
CALCULUS
INTEGRATION
,.
Consider same function 01 lxiy ) that exists in the Z plane
Where parametric equation
Ms
(x y) be
rlss
•
, can
expressed as a )
So we have 01 ( )
•
. \
x Z
y
lg
\ 1
.
New consider path C on the x -
y plane
which is parameterised by the variable It )
* We can new
integrate 01 an
the
path C
flx , y ) arz " " "
#I
*
Mt ) =
( xlt ) , ylt ) ) :
× (a)
←
,
.
yla )
f. #)
l.MY#lihikteEt:nmnnetYet
surface
x
t=b
Ib) ,
,
the
ylb )
beneath
area of
the
The
'
curtain
¢
'
b
fcads f HH ) dd÷
* =
0 at
a
, *
direction
*
a- b-
e.g. Integrate 01 =xiy2 from ) 11,0
to ( 0,1 )
along
the paths shown
'
.
*
for integration to work
(.cz
b- Co. 2) ° .
-
,
• C ,
the
it
,
is to
necessary express
, •
•
Cir parametric equations
'
, } as
c
y:
,
Q 2,0 )
Along as :
-
parametric farm
rlt ) =
a- + t( b- -
e) of a straight line
gip ;'
÷
"
pgaratmneterisf '
limits
:[6-
fcads
:
BHH )
fba ddttdt
=
b :c on )=a t.tt
ftp.tfgEY.jhmr#atuba
.
1
t
=
1
a :( 1,0 )
=
( l -
t ,t )
;':*
If
t O
III
Is
=
' "
antes
"
asayarattzn
=fz[t ¥+53 ]
1.
-
x= it
=f2[
2
( i tat
-
's ) -
( o )] Y2= A
rz ±
=
.
