Integration II: Partitioning the x-axis.

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Lesson Summary: 
We divide the $x$-axis between the limits of integration into an arbitrary set of closed intervals. We provide notation and use pictures to show several arbitrary partitions.
Lesson Inputs: 
Open and Closed Intervals
Sets and Set Notation
The Length of Intervals
Lesson Outputs: 
Integration IV: Definite Integrals As Riemann Sums
Partition Limits

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Lesson: 

This lesson is about setting up a notation we will use later to define a definite integral. In other words, we aren't doing math as much as we are defining words and math notation.

First, recall that a closed interval between $a$ and $b$ is the set of all values between $a$ and $b$ and including $a$ and $b$. In set notation we say that

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Partition
Assume that we are given the closed interval $[a,b]$. A partition of $[a,b]$ is a set of numbers with the following two properties:
  • The partition includes the numbers $a$ and $b$.
  • All numbers other than $a$ and $b$ are between $a$ and $b$. In other words, they are greater than $a$ and less than $b$.

For example, one possible partition of the closed interval <Sign in to see all the formulas> could be <Sign in to see all the formulas>. Note that this is a partition because the set includes the numbers $1.5$ and $10$ and all the other numbers are between them.

In general, our notation for a partition will be

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with the assumption that

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$P$ is a partition because it includes the numbers $a$ and $b$ and all the other numbers are between them.

The extremely important thing to understand is that we can "break" the $x$-axis between $a$ and $b$ into smaller, nonoverlapping closed intervals. In particular, we use the partition to describe $[a,b]$ as $N$, nonoverlapping closed intervals:

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and each of these closed intervals has a length. Our notation is that

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(Recall the use of the <Sign in to see all the formulas> symbol here.)

Because it will be used a lot in different proofs and definitions, we need to define the norm of the partition. We denote it by <Sign in to see all the formulas> and it is defined as

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So the norm of the partition is the length of the longest interval in the partition.

Finally, we see why we used the notation for a partition as <Sign in to see all the formulas> and not <Sign in to see all the formulas>: because the partition breaks $[a,b]$ into $N$ intervals and not $N+1$ intervals. There is nothing mathematical here. We just specified the notation of the partition so we could immediately see how many closed intervals we are breaking $[a,b]$ into.