DefiniteIntegralApplications

# Definite integrals

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, P_{n}, 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 P

_{n}, (lower area) s_{Pn}(f)- The higher sum of f associates to the partition P

_{n}, (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]**

# Definite integral

Approximation to a concept of definite integral

Creado con GeoGebra – Compartida por vmonterreal

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