## Problem on evaluating a line integral

Let $\mathbf{F}(x,y) = \langle 2, 3 \rangle$. Suppose $C$ is a curve connecting $(0,0)$ to $(1,1)$. Does the value of $\int_C \mathbf{F}\cdot d\mathbf{r}$ depend on the shape of the curve $C$? If not, find the value of the integral.
• ## Solution

We are being asked if the value of a line integral is path independent. Recall that
We identify $u(x,y) = 2$ and $v(x,y) = 3$.
Because $\partial_y u = \partial_x v = 0$, $\mathbf{F}$ is consevative. The value of the line integral does not depend on the shape of the path.

#### Finding the value of the integral

Recall that
We have already shown that $\bfF$ is conservative. Hence, we recall the fundamental theorem of line integrals:
To apply this theorem, we need to find the potential function $\phi(x,y)$.
Applying the fundamental theorem of line integrals, we can compute that $$\int_C \mathbf{F}\cdot d\mathbf{r} = \phi(1,1) - \phi(0,0) = 5.$$