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 ∪. 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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