How Can We Approximate Integrals Using Summations?

[ronm]

Dear all, I am struggling to understand a simple integration problem. Here is my integration:

Integration [0, t] {d S(u) / r(u)}

My friend says that above integration can be approximately written as

Sum [ti <= t] {si / ri}

S is the step function increasing by si at each step.

Could I explain my problem properly? Can somebody please explain me how I can write above integration as the summation approximately? It would be really good if you can explain it graphically. What is the Generally rule to handle this kind of scenario?

Appreciate your help.

 

[mathman]

http://en.wikipedia.org/wiki/Trapezoidal_rule

Start with this.