# Line integral path independence proof check

Find the work done by the force $$F(x, y, z) = (x^4y^5, x^3)$$ along the curve C given by the part of the graph of $$y$$ = $$(x^3)$$ from $$(0, 0)$$ to $$(-1, -1)$$.

I first checked for independence, which did not work.

Next I parameterized the curve by $$r(t) = [x(t),y(t)]$$, \begin{align*} x(t) &= t, \\ y(t)&=t^3 \end{align*} which has $$dr= (1, 3t^2).$$

Computing the work is then $$\int_0^{-1} (t^{19},t^3)\cdot (1,3t^2)\,dt = 11/20.$$ Is this correct?

Looks fine to me. By "check independence" I assume you mean that you computed that $$\nabla \times F \neq 0$$ so that $$F$$ has no chance of being integrable.
As a sanity check, when $$x<0,y<0$$ then $$F$$ points to the bottom-left, so that a positive work along the given curve segment is expected.