Relation between points and intermediate line connecting them

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If t represents time and v represents stock value, I have the following questions:

  1. What does the area under the curve represent?

  2. What does the product $v_i*L_{i-1}$ represent?

  3. What does $ \sum v_i*L_{i-1}$ ?

  4. Could using vectors help if we assume that the line segments are vectors?

What I am looking for is a representation or a meaning similar to that given such as $velocity=dx/dt$ and that Area under a velocity-time graph = distance traveled. Even a formula to represent the sum will be an answer.

I tried to find a logical meaning for what the above represent but could not find any!

This is not a homework.

Your help is appreciated. Thx.

Edit: On 3-Aug-16, Added question #4.

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