Curve Integrals

May 2010
2
0
"The integral I is defined by

I=∫F.dr where F(x,y)=(x-y),(xy)

And C is the triangle with vertices (0,0), (1,0) and (1,3) traversed anticlockwise.

Calculate the line integral directly"

I don't know how to get started on this! Please help?
 

matheagle

MHF Hall of Honor
Feb 2009
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1,146
I'd rather use Green's Theorem.
BUT if you wish to do this directly you need to figure out the three lines parametrically.

From (0,0) to (1,0)
you can let y=0, x=t where 0<t<1
note that dy=0, dx=dt

From (1,0) to (1,3)
you can let y=t, x=1 where 0<t<3
here dy=dt, dx=0

Finally you have
(1,3) to (0,0)
you can let y=3t, x=t BUT 1<t<0 since you are heading to (0,0)
then dy=3dt, dx=dt
 
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Reactions: NoEyedClare
May 2010
2
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The next part of the question is to verify using Green's Theorem. Thanks for that! I knew it involved parametric equations, just wasn't sure on how to set them up.