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.