The problem is:
Consider the vector field: $$\textbf{F}= 4x^3y^3 \,\textbf{i} + (1+3x^4y^2) \,\textbf{j}$$
a) Find a potential function $ϕ(x,y)$, i.e. a function $ϕ(x,y)$ such that $\nabla ϕ= \textbf{F}.$
b) Compute $\int_C \textbf{F} \cdot \mathrm d \textbf{r}$ over curve $C$ given by $$\textbf{r}(t)=\frac{\sin(t^2-t)}{2(\pi-1)} \,\textbf{i} + \frac{t}{\pi} \,\textbf{j}, \quad t \in (0, \pi)$$
I found the function for part (a) as $$\phi(x, y) = x^4y^3+y+C$$ but I didn't know what to do once I came to part (b). Am I supposed to take the derivative for $\textbf{r}(t)$?
Hint: Given $\phi(x,y)$ such that $\nabla \phi = F$, and a curve $C$ with endpoints $(x_{1},y_{1}), (x_{2},y_{2})$ the fundamental theorem for line integrals tells you
$$\int_{C} F\cdot dr = \phi(x_{2},y_{2})-\phi(x_{1},y_{2})$$
You found $\phi$, so you need to compute $(x_{1},y_{1})$ and $(x_{2}, y_{2})$ for the curve parametrized by $t$. What do you get when you compute $r(0)$ and $r(\pi)$?