When Green's Theorem Creates An Integral Of $0$, What Happens?

Question

"Evaluate the counter-clockwise integral:

$\int_c 4x^3ydx + x^4dy$

for any closed path $C$."

My Work

This is obviously a job for Green's theorem, where $P = 4x^3y$ and $Q = x^4$

$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 4x^3 - 4x^3 = 0$

Therefore:

$\int_c 4x^3ydx + x^4dy = \int\int_D 0dxdy$

So...what exactly does this mean? I don't know how to interpret what the representation of this integral is, or how it can be evaluated for any legal $C.$

Answer 1

It means the integral around any closed path is zero. This is a consequence of your vector field $(4x^3y,x^4)$ being conservative. Can you find the potential field it is the gradient of?