Setting up line integral of a vector field

Question

Since a line integral of a vector field is defined as follow:

For a problem like this, how do i set up the integral

Show details to find the work done by the force field $\mathbf{F} =(2x+y)\,\mathbf{i}+x\,\mathbf{j}$ in moving an object from $P(1, 1)$ to $Q(4, 3).$

My question is:

1. how to set up the integral for this

   1.1 how to setup the bounds (a,b) (what are they)

   1.2 what is r(t) and how to find it in this case

Answer 1

You need a vector function that goes over the line for a bound you define, so there's infinite choices. I find it easiest to keep the bound between $[0, 1]$. Our function could be $r(t) = (1 + 3t)i + (1 + 2t)j, 0 < t < 1$

Edit: I misread! They have not specified the path. Instead, one has to recognize the field is conservative, and thus the path $r(t)$ does not matter.

The potential function is given by $Φ(x, y) = x^2 + xy$. If you take the gradient of that, you get $f$. Then, the line integral is given by $Φ(4,3) - Φ(1,1) = 26$

Note that because the path does not matter, we could have carried on with a line and have gotten the same answer.