How do I use the fundamental theorem of line integrals to evaluate the integral
(3x-y+1)dx - (x+4y+2)dy from (-1,2) to (0,1)?
*sigh* i know i responded to similar questions here, but i can only find the ones with 3 variables, not two, so they might confuse you. i can't find a way to give you hints for this other than do the problem...or most of it anyway. other than that, i will tell you to go read your textbook. i know they gave an example of how to do this
we want to find a function $\displaystyle f(x,y)$ so that $\displaystyle f_x = 3x - y + 1$ and $\displaystyle f_y = -x - 4y - 2$
(we know we can do this since the "equations" are exact: $\displaystyle \frac {\partial }{\partial y}(3x - y + 1) = \frac {\partial }{\partial x}(-x - 4y - 2)$)
so take $\displaystyle f_x = 3x - y + 1$, then
$\displaystyle f(x,y) = \int f_x~dx = \frac 32x^2 - xy + x + g(y)$
(our constant of integration is a function of y, since this will die when we differentiate with respect to x)
$\displaystyle \Rightarrow f_y = \frac {\partial }{\partial y}f(x,y) = -x + g'(y)$
But $\displaystyle f_y = -x - 4y - 2$, so we have $\displaystyle -x -4y - 2 = -x + g'(y)$. hence
$\displaystyle g'(y) = -4y - 2$
$\displaystyle \Rightarrow g(y) = -2y^2 - 2y + K$
Therefore, $\displaystyle f(x,y) = \frac 32x^2 - xy + x -2y^2 - 2y + K$ is the (potential) function we seek