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Definite integral

DEFINITE INTEGRAL

We are going to calculate the area under the graph of a function in an interval, the area of R.
To get it, we do a partition, Pn, of the interval [a,b] in n subintervals:

  a = x0<x1<x2<……<xn = b

Then, we have two options to calculate the area:
•The lower sum of f associates to the partition Pn, (lower area) sPn(f)
•The higher sum of f associates to the partition Pn, (higher area) SPn(f)
Obviously:                            sPn(f)≤ area (R) ≤ SPn(f)
If we choose another partition, Pn’, n’ > n, then:
  
         sPn(f)≤ sPn’(f)≤ area (R) ≤ SPn’(f) ≤ SPn(f)
If we do the limits as n approaches ∞ and they are equal, then:

    this is called definite integral of f between a and b, and it is said that f is integrable in [a,b]