Path Integral problems
Have a problem that i have had a go at but am completely unsure if am anywhere near on the right track any help would be appreciated.
Evaluate the path integral;
Where f = xy;
And the path C is the part of the circle centred at the origin starting at (0,2) going anticlockwise ending at (-2,0).
I did that x=2cost, y=2sint;
so that c(t) = (2cost , 2sint),
where 0< t < pi/2
but am not sure i have set this up properly am a bit lost here.
Well, one thing that's a bit unclear is that you have , yet you are only parameterizing two variables.
Originally Posted by monster
I think you want
Actually, I think the first problem is in the limits for t. As you go from (0,2) to (-2,0), that is, with , then .
Next , and , so .
And since this is the line integral of a scalar function, ds is the arc length parameter:
, so in terms of t,
and you have
Originally Posted by redsoxfan325
Where are you getting the z term, redsoxfan325? This is a two-dimensional problem (in the xy plane), so only x and y are involved; there's no z.
He/she said , but it doesn't change the answer either way because .
Originally Posted by TwistedOne151
c(t) = (2cos(t) , 2sin(t))
so c'(t) = ( - 2sin(t), 2cos(t))
so [c'(t)] = 2
then made sub u = cost to give
Is the way i have gone about it correct?
and thank you for all help guys.
Yeah, you did it correctly.