Let \(\mathbf{F}(x, y, z)=\left\langle z^{2}, x, yightangle\), and let \(C\) be the curve that is given by \(\mathbf{r}(t)=\left\langle 3+5 t^{2}, 3-t^{2}, tightangle\) for \(0 \leq t \leq 2\).

(a) Calculate \(\mathbf{F}(\mathbf{r}(t))\) and \(d \mathbf{r}=\mathbf{r}^{\prime}(t) d t\).

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