The force function $\displaystyle \bold{F} = -\bold{\hat{x}}kx - \bold{\hat{y}}ky$ is using Cartesian coordinates.

Find the work done from (1,1) to (4,4) using the following path:

$\displaystyle (1,1) \to (1,4) \to (4,4)$

So, what I did was:

$\displaystyle \int^{4,4}_{1,1} \bold{F}\cdot ~d\bold{r}$

$\displaystyle = \int^{4,1}_{1,1} \bold{F}\cdot ~d\bold{r} + \int^{4,4}_{4,1} \bold{F}\cdot ~d\bold{r}$

$\displaystyle = \int^{4,1}_{1,1} (-kx ~dx - ky ~dy) + \int^{4,4}_{4,1} (-kx ~dx - ky ~dy)$

$\displaystyle = \int^4_1 (-kx ~dx) + \int^1_1 (-ky ~dy) + \int^4_4 (-kx ~dx) + \int^4_1 (-ky ~dy)$

$\displaystyle = -k\left(\int^4_1 x ~dx + \int^1_1 y ~dy + \int^4_4 x ~dx + \int^4_1 y ~dy\right)$

$\displaystyle = -k\left[\left(8 - \frac{1}{2}\right) + \left(8 - \frac{1}{2}\right)\right]$

$\displaystyle = -k(15) = -15k$

Is this right?