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

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24 Nov 2015
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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
π‘₯∈[π‘Ž,𝑏]
Let π‘›πœ–!=1,2,3,4…=π‘›π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™!π‘›π‘’π‘šπ‘π‘’π‘Ÿπ‘ 
𝐿𝑒𝑑!βˆ†π‘₯=
(
π‘βˆ’π‘Ž
)
𝑛
=Sub-interval width
As β€˜n’ gets larger and larger βˆ†π‘₯ gets
smaller and smaller
lim
!β†’!
βˆ†π‘₯=0
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Right endpoint of the kth sub-interval is π‘₯!=π‘Ž+π‘˜βˆ†π‘₯=π‘Ž,π‘€β„Žπ‘’π‘Ÿπ‘’!π‘˜=0!
π‘₯!=π‘Ž+π‘›βˆ†π‘₯=𝑏,π‘€β„Žπ‘’π‘Ÿπ‘’!π‘˜=𝑛
In the nth Riemann Sum is defined as follows:
𝑆𝑛 =𝑓π‘₯1βˆ†π‘₯+𝑓π‘₯2βˆ†π‘₯+𝑓π‘₯3βˆ†π‘₯+β‹―+𝑓π‘₯𝑛 βˆ†π‘₯
In the Summation Notation (Sigma Notation)
𝑆𝑛 =π‘“π‘˜βˆ†π‘₯
!
!!!
** β€˜k’ must start at 1
Definition: The definite integral of y=f(x) from β€˜a’ to β€˜b’ is:
𝑓π‘₯𝑑π‘₯
!
!
=lim
!β†’!𝑓π‘₯!βˆ†π‘₯
!
!!!
𝑓π‘₯𝑑π‘₯ =#
!
! is a special real number that is the limit of a certain kind of sum
Special, Important Case:
If 𝑓π‘₯β‰₯0 when π‘₯βˆˆπ‘Ž,𝑏 then,
𝑓π‘₯𝑑π‘₯
!
!
=!π‘‘β„Žπ‘’!π‘Žπ‘Ÿπ‘’π‘Ž!π‘œπ‘“!π‘‘β„Žπ‘’!π‘Ÿπ‘’π‘”π‘–π‘œπ‘›!π‘’π‘›π‘‘π‘’π‘Ÿ!π‘‘β„Žπ‘’!π‘π‘’π‘Ÿπ‘£π‘’,π‘Žπ‘π‘œπ‘£π‘’!π‘₯βˆ’π‘Žπ‘₯𝑖𝑠!π‘€β„Žπ‘’π‘Ÿπ‘’!π‘Žβ‰€π‘₯≀𝑏
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