Solution
= Solution
For $\mathbf r(t)=(t^2,t)^T$, the <tangent vector> is $(2t,1)^T$ and $\mathbf F(\mathbf r(t))=(t^3,t^4+t)^T$. Thus
$$
\int_0^1(2t^4+t^4+t)\,dt=\frac35+\frac12=\boxed{\frac{11}{10}}.
$$
The unequal values for the same endpoints already rule out a <conservative vector field>.