Let \(\mathbf{F}=\left\langle y^{2}, x^{2}ightangle\), and let \(C\) be the curve \(y=x^{-1}\) for \(1 \leq x \leq 2\), oriented from left to right.

(a) Calculate \(\mathbf{F}(\mathbf{r}(t))\) and \(d \mathbf{r}=\mathbf{r}^{\prime}(t) d t\) for the parametrization of \(C\) given by \(\mathbf{r}(t)=\left\langle t, t^{-1}ightangle\).

(b) Calculate the dot product \(\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}^{\prime}(t) d t\) and evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).