This chapter includes notes on Line Integration, Path Integral in Vector Fields, Double and Repeated Integrals, Volume between Two Surfaces and Green's Theorem. Each section includes multiple worked examples. Notes were taken on an iPad
,.
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 ±
=
.
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