9. (10 pts) Let $C$ be the curve in $\mathbb{R}^3$ defined by the parametric equation

$$x(t)=\cos(t),\ y(t)=\sin(t),\ z(t)=t$$

for $0 \le t \le a$. Suppose that the arc length of $C$ is $\dfrac{\sqrt{2}\pi}{4}$. Evaluate the line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ of the vector field $\mathbf{F}=xz\mathbf{i}+yz\mathbf{j}+x^3\mathbf{k}$ on $\mathbb{R}^3$.