I have a triangular area defined by $(x_1, y_1), (x_2, y_2), (x_3, y_3) $ (assume they are arranged in the anti-clockwise direction).
And I have a function $T(x, y)$ defined by:
$$T(x,y) = \begin{cases} T_0, &\text{ if }\, T_0 > r(x,y)\\ r(x,y), &\text{ else}\end{cases}$$
Here $r(x, y)$ is a planar function, i.e., $r(x, y) = ax + by + c$ and $T_0$ is a constant.
Now I want to decompose the surface integral of $T(x, y)$ over the triangular area into some combinations of its line integral on three edges.
How I solve this problem in an analytical way?