Solution
= Solution
Parametrize the straight path by $\mathbf r(t)=(t,t,t)$, $0\leq t\leq\pi$. Then $\mathbf F_3\cdot d\mathbf r=(4t^3+\sin t)dt$, so direct evaluation of the <line integral> gives
$$
\boxed{\int_C\mathbf F_3\cdot d\mathbf r=\int_0^\pi(4t^3+\sin t)dt=\pi^4+2.}
$$
The <fundamental theorem for line integrals> supplies an independent check: this is $\phi_3(\pi,\pi,\pi)-\phi_3(0,0,0)$.