Height ( ℎ 𝑖 h i ​ ): The height of each trapezoid can be interpreted as the average of the two y-values (since the trapezoid is bounded by these values and the x-axis). For the 𝑖 i-th trapezoid, we can compute it as:

ℎ 𝑖

𝑦 𝑖 + 𝑦 𝑖 − 1 2 h i ​

2 y i ​ +y i−1 ​

​

Width ( 𝑤 𝑖 w i ​ ): The width is simply the difference between the x-coordinates of the points:

𝑤 𝑖

𝑥 𝑖 − 𝑥 𝑖 − 1 w i ​ =x i ​ −x i−1 ​

Area computation: Finally, the AUC (the sum of the areas of all the trapezoids) is given by:

𝐴 𝑈 𝐶

∑ 𝑖

1 𝑛 ℎ 𝑖 ⋅ 𝑤 𝑖 AUC= i=1 ∑ n ​ h i ​ ⋅w i ​

Where each ℎ 𝑖 h i ​ is the average of the y-values at 𝑖 i-th and ( 𝑖 − 1 ) (i−1)-th points, and 𝑤 𝑖 w i ​ is the distance between consecutive x-coordinates.