# MATA32 Lecture 21: The Definite Integral and the Fundamental Theorem of Calculus

MATA32 – Lecture 21: The Definite Integral and the Fundamental Theorem of

Calculus

The Definite Integral

We’ll see that the definition of the definite integral is based on these simple ideas

Begin with a given interval [a,b] (a<b) and a continuous MATA32 function y= f(x) where

𝑥∈[𝑎,𝑏]

Let 𝑛𝜖!=1,2,3,4…=𝑛𝑎𝑡𝑢𝑟𝑎𝑙!𝑛𝑢𝑚𝑏𝑒𝑟𝑠

𝐿𝑒𝑡!∆𝑥=

(

𝑏−𝑎

)

𝑛

=Sub-interval width

As ‘n’ gets larger and larger ∆𝑥 gets

smaller and smaller

lim

!→!

∆𝑥=0

Right endpoint of the kth sub-interval is 𝑥!=𝑎+𝑘∆𝑥=𝑎,𝑤ℎ𝑒𝑟𝑒!𝑘=0!

𝑥!=𝑎+𝑛∆𝑥=𝑏,𝑤ℎ𝑒𝑟𝑒!𝑘=𝑛

In the nth Riemann Sum is defined as follows:

𝑆𝑛 =𝑓𝑥1∆𝑥+𝑓𝑥2∆𝑥+𝑓𝑥3∆𝑥+⋯+𝑓𝑥𝑛 ∆𝑥

In the Summation Notation (Sigma Notation)

𝑆𝑛 =𝑓𝑘∆𝑥

!

!!!

** ‘k’ must start at 1

Definition: The definite integral of y=f(x) from ‘a’ to ‘b’ is:

𝑓𝑥𝑑𝑥

!

!

=lim

!→!𝑓𝑥!∆𝑥

!

!!!

𝑓𝑥𝑑𝑥 =#

!

! is a special real number that is the limit of a certain kind of sum

Special, Important Case:

If 𝑓𝑥≥0 when 𝑥∈𝑎,𝑏 then,

𝑓𝑥𝑑𝑥

!

!

=!𝑡ℎ𝑒!𝑎𝑟𝑒𝑎!𝑜𝑓!𝑡ℎ𝑒!𝑟𝑒𝑔𝑖𝑜𝑛!𝑢𝑛𝑑𝑒𝑟!𝑡ℎ𝑒!𝑐𝑢𝑟𝑣𝑒,𝑎𝑏𝑜𝑣𝑒!𝑥−𝑎𝑥𝑖𝑠!𝑤ℎ𝑒𝑟𝑒!𝑎≤𝑥≤𝑏