Calculating an Intergral over C Using Force F and Circle C

[jlmac2001]

Find the work (intergral over C )F dot dr done by a force F=yi + xj in going all the wy counterclockwise around circle C give by x^2+y^2+2x=0, by the easiet technique you know.

Would i get a double intergral over C (-1) dxdy? How would I get C?

 

[himanshu121]

U can interchange dy into dx and vice versa from the equation of circle and it will be easy to integrate

 

[HallsofIvy]

The easiest way is this: Since d(y)/y= 1= d(x)/dx, this is a "conservative" force field (mathematically, ydx+ xdy is an "exact differential") and so its integral around any closed path is 0.

I'm not sure where you got "-1" from. Using Green's theorem the integrand would be d(x)/dx- d(y)/dy= 1- 1= 0 just as above.

Saying "How would I get C?" makes it sound as if you think C is a constant. You don't have to "get" C: C is the path given.

IF the problem were to integrate, say, ydx+ 3xdy, then we would integrate $$\int (\frac{d(3x)}{dx}-\frac{d(y)}{dy})dA$$
= $$2\int dA$$ which is just 2 times the area of the circle.