We can use a plane sweep method sweeping from left to right. There are three kinds of event points for the algorithm (denoted in Figure 1 by gray arrows):

• beginning of new triangle (value ~x_i~ from description);
• crossing point of one triangle horizontal edge with others hypotenuse;
• rightmost vertex of some triangle (value ~x_i+m_i~ from task description).

If we look at these event points consecutively, we can calculate common area increasing, if we at each event point ~x_k~ know number ~n_k~ of actual line segments and sum of their lengths ~s_k = \sum \Delta y_i~, where ~\Delta y_i~ is the length of one segment. Example with three segments are shown in Figure 2.

Then increment for each gap between two neighbour event points ~(\Delta x = x_{k+1}-x_k)~ can be calculated as ~s_k \Delta x - n_k \frac{(\Delta x)^2}{2}~. Performing sweeping from left to right and summing up these increments we finally obtain total common area of all triangles.