See attachment for details of integration.
To answer your second question have you studied Path Independence Yet?
Let the field be F = f i + g j
if df/dy = dg/dx F is the gradient of a scalar function phi.
The line integral is then simply phi(endpoint) - phi(initialpoint)
If you haven't studied this you will including how to find phi -- note for the phi I gave grad(phi) = F
and phi(1,1)- phi(0,0) = 3
For a complete discussion of path independence and conservative fields
see my web site Line Integrals