Part 1: Line Integrals over Parameterized Curves

One of the most important tools for studying vector fields is the line integral, where the word "line" should be interpreted here with its more general definition as ``an ordered succession of points.'' That is, a line integral is an integral over a curve which has a well-defined orientation.

Specifically, if C is a curve in the xy-plane parametrized by r( t) = á x( t) ,y( t) ñ for t in [ a,b] and if F(x,y) = á M( x,y) ,N( x,y) ñ , then the line integral of F(x,y) over C is defined

ó õ C  M( x,y) dx+N( x,y) dy = ó õ b a  M(x( t) ,y( t) )  dx dt +N( x(t) ,y( t) )  dy dt   dt

(1)

Thus, line integrals are evaluated by ``pulling back'' to an ordinary integral in the parameter used in parameterizing the curve.       

EXAMPLE 1    Evaluate

ó õ C  ydx-xdy

over the curve C parametrized by r( t) = ácos( t) ,sin( t) ñ for t in [ 0,p] .

   

Solution: According to (1), the line integral becomes

ó õ C  ydx-xdy = ó õ p 0  æ è y  dx dt -x  dy dt ö ø dt

Since x = cos( t) and y = sin( t) , we have dx/dt = -sin( t) and dy/dt = cos(t) , so that

ó õ C  ydx-xdy = ó õ p 0  ( sin( t) [ -sin( t) ] -cos( t) [ cos( t)] ) dt = - ó õ p 0  sin2( t) +cos2( t) dt = -p

Similarly, if C is a curve in R³ parameterized by r( t) = á x( t) ,y( t),z( t) ñ , then

ó õ C  Mdx+Ndy+Pdz = ó õ b a  æ è M  dx dt +N  dy dt +P  dz dt ö ø dt

(2)

is the line integral of F = á M,N,P ñ over C.       

EXAMPLE 2    Evaluate

ó õ C  xdx+ydy+z2dz

when C is the helix parameterized by r( t) = á cos( t) ,sin( t) ,t ñ for t in [ 0,2p] .       

Solution: To do so, we notice that

ó õ C  xdx+ydy+z2dz = ó õ 2p 0  æ è x  dx dt +y  dy dt +z2  dz dt ö ø dt

However, x = cos( t) , y = sin( t) , and z = t implies that

ó õ C  xdx+ydy+z2dz = ó õ 2p 0  ( cos( t)[ -sin( t) ] +sin( t) [ cos( t) ] +t2[ 1] ) dt = ó õ 2p 0  -sin( t) cos( t) +sin(t) cos( t) +t2dt =  t3 3 ê ê 2p 0  =  8p3 3