Solution
= Solution
{c}
A <scalar potential> is $U(x,y)=x^3y+y^4$, since $\nabla U=(3x^2y,x^3+4y^3)$. The <fundamental theorem for line integrals> therefore reduces the <line integral> to its endpoints. The curve starts at $(0,1)$ and ends at $(2,1)$, so
$$
\boxed{\int_C\mathbf F\cdot d\mathbf r=U(2,1)-U(0,1)=9-1=8.}
$$