Problem:

Evaluate the line integral of the vector field $\mathbf{F}(x, y, z) = y\mathbf{i} + xz\mathbf{j} + z^2\mathbf{k}$ along the curve $C$, where $C$ is given by the parametric equations: $\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k}, \quad t \in [0, 1].$

Solution Steps:

1. Find $\mathbf{r}'(t)$: Differentiate $\mathbf{r}(t)$ with respect to $t$.
2. Evaluate $\mathbf{F}(\mathbf{r}(t))$: Substitute the parametric equations into $\mathbf{F}(x, y, z)$.
3. Compute the dot product $\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)$.
4. Integrate the result with respect to $t$ over $[0, 1]$.

Would you like me to work out the solution or provide hints for each step?