AUC per time is nothing else than the mean? Using AUC is in vogue and has found his place also in clinical research (example). What I don't understand is AUC per time. For example, if a clinical or psychological parameter is measured over time. My assumption is:
AUC per time using trapezium rule is nothing else than the mean of the measurements.  
If this is the case, why not just use mean?
 A: Suppose the measurements are $y_1, \dots, y_{n+1}$ taken at times $a = t_1 < t_2 < \cdots < t_n < t_{n+1}= b$.
Then the area under the curve (AUC) using the trapezoidal rule  is given by
$$
\text{AUC} = \frac12 \sum_{i=1}^{n} (t_{i+1}-t_i) (y_{i+1} + y_i).
$$
First, suppose that the intervals are equal.  Then for each $i$ we would have 
$$
t_{i+1} - t_i = \frac {b-a} {n}.
$$
This is because even though we have $n+1$ points, there are only $n$ intervals.
Then, when we normalize the AUC by $b-a$, we would would have
$$
\frac{\text{AUC}}{b-a} = \frac12 \sum_{i=1}^{n} \frac{1}{n} (y_{i+1} + y_i)
=\frac{1}{2n}y_1 +  \frac{1}{n}\sum_{i=2}^{n}y_i + \frac{1}{2n}y_{n+1}.
$$
So, intuitively, even though it seems as though if the intervals were equally spaced then we would see something like the mean, we are actually a little bit off.  It could equal the mean --- for example, consider the case where all the $y_i$ are equal to 0. 
Taking a closer look, we see that
$$
\begin{align}
\text{AUC} & =
\frac12\sum_{i=1}^n (t_{i+1}-t_i) y_{i+1} + \frac12 \sum_{i=1}^n (t_{i+1}-t_i)y_i \\
 &=\frac12 (t_2-t_1)y_1 + \frac12 (t_{n+1}-t_n)y_{n+1} +
\frac12\sum_{i=2}^{n} (t_{i+1}-t_{i-1}) y_{i}. 
\end{align}
$$
Normalizing by $b-a$ again, we get
$$
\begin{align}
\frac{\text{AUC}}{b-a} & =
 &=\frac{(t_2-t_1)}{2(b-a)}y_1 + \frac{(t_{n+1}-t_n)}{2(b-a)}y_{n+1} +
\frac12\sum_{i=2}^{n} \frac{(t_{i+1}-t_{i-1})}{(b-a)} y_{i}. 
\end{align}
$$

AUC per time using trapezium rule is nothing else than the mean of the measurements.

Not necessarily, I think.  Again, it is possible that this could equal the mean, but it does not have to be the case.
Intuitively, you can see that data points that are taken in the middle of large time intervals are weighted more heavily than other points.   Points that are closer together in time receive less weight.
