Let \(\mathbf{F}=abla f\), and determine directly \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each of the two paths given, showing that they both give the same answer, which is \(f(Q)-f(P)\).

\(f=x^{2} y-z, \mathbf{r}_{1}=\langle t, t, 0angle\) for \(0 \leq t \leq 1\), and \(\mathbf{r}_{2}=\left\langle t, t^{2}, 0ightangle\) for \(0 \leq t \leq 1\)