Solution
= Solution
On the first segment only $z$ varies and $F_z=x^2y=0$. On the second only $y$ varies and $F_y=x^2z=0$. On the last segment $y=z=\pi$, so the <line integral> is
$$
\boxed{0+0+\int_0^\pi(2\pi^2x+\sin x)dx=\pi^4+2.}
$$
The agreement is the expected <path independence> of a <conservative vector field>.