Calculus Chapter 4

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 ±
=
.