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 = x_{0}<x_{1}<x_{2}<……<x_{n} = b

Then, we have two options to calculate the area:

•The lower sum of f associates to the partition Pn, (lower area) s

_{Pn(f)}•The higher sum of f associates to the partition Pn, (higher area) S

_{Pn(f)}Obviously: s

_{Pn(f)}≤ area (R) ≤ S_{Pn(f)}If we choose another partition, P

_{n’}, n’ > n, then: s

_{Pn(f)}≤ s_{Pn’(f)}≤ area (R) ≤ S_{Pn’(f)}≤ S_{Pn(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]