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I believe that to solve this problem, you must set up two separate parametrizations and two separate integrals. The first parametrization/integral should be the line segment from (1, 1) to (3, 1) and the second parametrization/integral should be from (3, 1) to (3, 6).

To parametrize these lines, you find the differences in $x$ and $y$ between them and use them in your direction vector. Typically $t$ will range from $0$ to $1$ (see Christina's answer for an explanation why).

The first line is parametrized by:

$$ \vec p_1 (t) = \langle 1, 1 \rangle + \langle 2, 0 \rangle, $$

where $0 \leq t \leq 1$.

The second line can be parametrized in a very similar fashion. Then you must add the result of the two definite integrals together to get the total integral.

My apologies for the terse answer, I am on my phone right now!