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 \(f(Q)-f(P)\).

\(f=z y+x y+x z, \mathbf{r}_{1}=\langle t, t, tangle\) for \(0 \leq t \leq 1, \mathbf{r}_{2}=\left\langle t, t^{2}, t^{3}ightangle\) for \(0 \leq t \leq 1\)