Integration a 2D Gaussian distribution over a line, what is the meaning of that and how to achieve that?

Question

my problem is to find the best error model for a sensor. I got measurement(points) and reference surface(line segments) and I want a error model(2D gaussian distribustion) which gives the maximum likelihood. What I did is that I integrate the 2D gaussian distribution over a line segment$$ \int_{ \mathbf{p_1}}^{\mathbf{p_2}} \frac{1}{2 \pi \sqrt{ det \Sigma } } e^{- \frac{1}{2} (\mathbf{p}-\mathbf{ \mu })^T \Sigma^{-1}(\mathbf{p}-\mathbf{ \mu }) } \, d \mathbf{p} $$ where $\mathbf{p} = \left[\begin{matrix} x \\ y \end{matrix}\right] $ is the point on a reference line, $\mu$ is the measurement point and $\Sigma$ is the covariance matrix of the error model(2d Gaussian).
My first question would be does this even make sense to integrate over a line?
The second question is that since $\mathbf{p}$ is on a line, I want to map $\mathbf{p} \in (\mathbf{p_1}, \mathbf{p_2})$ to t$\in(0, 1)$. Then the Integral would be $$ \int_{0}^{1} P(\mathbf{p(t)}) \mathbf{p}^{'} (t) \, dt $$ what is this $\mathbf{p}^{'} (t)$? should it be the distance from $\mathbf{p_1}$ to $\mathbf{p_2}$? what if I need to integral over $t \in (0.2, 0.8)$, does this $\mathbf{p}^{'} (t)$ also change, since the integral intervl changes?
I would really appreciate that if someone can help me...