Line integral of F=(6x²y², 4x³y+5y⁴) on a first quadrant boundary curve

[Kuma]

Homework Statement

Trying to evaluate the following line integral:

integral F ds where F = (6(x^2)(y^2), 4(x^3)(y) + 5y^4)
and the path is the boundary curve of the first quadrant below y = 1-x^2 in a clockwise direction.

Homework Equations

The Attempt at a Solution

So since the curve is piecewise smooth closed simple and closed curve I can use greens theorem. Simply put I get an answer as 0 since dF1/dy = dF2/dx. Is that right?

 

[Dick]

Homework Statement

Trying to evaluate the following line integral:

integral F ds where F = (6(x^2)(y^2), 4(x^3)(y) + 5y^4)
and the path is the boundary curve of the first quadrant below y = 1-x^2 in a clockwise direction.

Homework Equations

The Attempt at a Solution

So since the curve is piecewise smooth closed simple and closed curve I can use greens theorem. Simply put I get an answer as 0 since dF1/dy = dF2/dx. Is that right?

I would agree with that.

 

[wangchong]

Your vector field is conservative. So the line integral would be zero over any closed curve (even non-simple closed curve!). A vector field is conservative if it is the gradient of a potential function.