"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 $C.$
Best Answer
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?