Hey, I'm trying to solve the following double line integral:

$\displaystyle
\int_{L_1}\int_{L_2} f(\textbf{x},\textbf{y})dsdl,
$

where $\displaystyle ds$ and $\displaystyle dl$ denote the line elements and the vector-valued function $\displaystyle f:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb {R}$ is given by

$\displaystyle
f(\textbf{x},\textbf{y})=\max\{0,(1-||\textbf{x}-\textbf{y}||_2)^2\},
$

where $\displaystyle ||\cdot||_2$ denotes the Euclidean norm.


I tried various things: Assuming that with the help of the angle $\displaystyle \theta_i$ and the radius $\displaystyle t_i$ the line $\displaystyle L_i$ can be parametrized then the integral could be rewritten to

$\displaystyle
\int^{l=+\infty}_{l=-\infty}\int^{s=+\infty}_{s=-\infty} f\left(\begin{pmatrix}t_1\cos\theta_1-s\sin\theta_1 \\ t_1\sin\theta_1+s\cos\theta_1 \end{pmatrix},\begin{pmatrix}t_2\cos\theta_2-s\sin\theta_2 \\ t_2\sin\theta_2+s\cos\theta_2\end{pmatrix}\right)d sdl.
$

Since the function is zero when $\displaystyle x$ and $\displaystyle y$ get too large, this expression should be well defined. However, I'm having a hard time to find the correct integrations limits.

Also, I think that there might not exist an analytic expression for the double integral. If that's the case, I would be happy if I could do at least one analytic integration (and use for the other integral some numerical integration scheme).

Does anyone have an idea? Any comment is hightly appreciated!

Many thanks,
DeeAna