Solution
= Solution
Parametrize the straight path by $\mathbf r(t)=(t,t,2t)$ for $0\leq t\leq1$. The conservative field along it is $(6t^5,4t^5,t^5)$ and the tangent is $(1,1,2)$. Thus
$$
\boxed{\int_A^B\mathbf F_1\cdot d\mathbf r=\int_0^1 12t^5\,dt=2}.
$$
The associated potential has $\phi_1(B)-\phi_1(A)=2$, in agreement with the <fundamental theorem for line integrals>.