 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

A <parametrization> of the straight <line segment> is $\mathbf r(t)=(t,t,t)$, $0\le t\le1$. Then $\mathbf F(\mathbf r(t))=(t,t^2,t)$ and $\mathbf r'(t)=(1,1,1)$. By the <line integral> formula,
$$
\boxed{\int_C\mathbf F\cdot d\mathbf r
=\int_0^1(2t+t^2)\,dt=\frac43.}
$$

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