Is Every Line Integral Zero with Green's Theorem?

[EV33]

Homework Statement

Use Green's Theorem to evaluate the line integralalone the given positvely oriented curve.

∫$$_{c}$$ sin(y)dx+xcos(y)dy, C is the ellipse x²+xy+y²=4

Homework Equations

The Attempt at a Solution

∫∫(cos(y)-cos(y))dA=∫∫0dA

Because this ends up being the double integral of zero, does this just mean my answer is zero?

 

[Dick]

That's right. The line integral around any closed curve will be 0.

 

[vela]

Yup.

 

[EV33]

Awesome. Thank you both.

 

[gomunkul51]

Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.

 

[Dick]

Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.

Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?

 

[gomunkul51]

Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?

It's a subtle point, I wanted EV33 to know that :)
It is very easy to make that mistake and think that any close line integral is zero.