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Let $X\geqslant 0$ be a random variable. Then, we have

$$\mathcal{E}(X)=\int_0^\infty P(X>t)dt$$

(provided $\mathcal{E}(X)$ exists).

Suppose we have a finite data set $\{(d_1, a_1), \ldots, (d_n, a_n)\}$ consisting of pairs or real numbers where $d_i$ stands for a level (height) of some vessel and $a_i$ is the area of the surface of the vesel at level $d_i$.

How can I apply the above mentioned formula to calculate the (expected) capacity of the vessel?

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A single vessel? Anything known about the shape? If the vessel is a conical cup, the measurements can tell us nothing about capacity. The cup could be very tall, but our measurements might involve only small amounts of water. – André Nicolas May 3 '12 at 16:12
The vessel is a lake, essentially. – Rumburak May 3 '12 at 16:53
A circular lake. – Rumburak May 3 '12 at 20:31

If $(d_i)$ is nondecreasing, try $$ V=\sum\limits_{i=1}^{n-1}\frac{a_i+a_{i+1}}2(d_{i+1}-d_i). $$ This is as nonsensical as several other equivalent formulas, but not more.

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