The Length of Intervals

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Lesson Summary: 
The length of the intervals $[a,b]$, $[a,b)$, $(a,b]$, and $(a,b)$ are all $b-a$.
Lesson Inputs: 
Sets and Set Notation
Open and Closed Intervals
Lesson Outputs: 
Integration II: Partitioning the x-axis.

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Lesson: 

It is beyond the scope of a beginning Calculus class to explain exactly and logically why a single point on the number line has a length of zero. However, with this concept we can motivate the folllowing four "rules":

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First, let's just assume that the length of the closed interval $[a,b]$ is $b-a$. This is hardly an assumption since we learned in elementary school that the distance between numbers $a$ and $b$ on the number line is $b-a$.

Now let's try to find the length of the interval of $(a,b]$. The method for adding an element to a set is called a union and has the notation $\cup$. For example, if we had the set <Sign in to see all the formulas> and unioned it to the set <Sign in to see all the formulas> then we would end up with a third set <Sign in to see all the formulas>. We would write <Sign in to see all the formulas> and say "C equals A union B."

Now let's assume that if <Sign in to see all the formulas> then

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Now since <Sign in to see all the formulas> then

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Since we are assuming that the length of $\{a\}$ is zero then

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We can make similar arguments for $[a,b)$ and $(a,b)$.

To summarize, we have not proven anything. It is beyond the scope of a beginning Calculus student to be able to prove these concepts. However, we hope to have motivated the following rules:

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