The Citizen Scientist

25 February 2005

How do we Use Limits?

by George E. Hrabovsky, President, MAST

Where We Have Been

Over the last several issues we have explored the ideas of sequences and limits. In this issue we will explore integration again.

Session 6: How do we use limits?

Now that we know what a limit is, it is natural to ask how we would use such a thing. Recall our definition of the definite integral,

If we think about this for a while, then we will realize that for the maximum of to tend toward zero then the number of vertical slices represented by the interval must be infinite, as we will see below. We write this as,

We can interpret this to mean that on the interval from to , we choose subintervals of width and choose a point within each subinterval. Let us assume that we must find the integral,

It is not possible to just use this without defining the points within the subintervals and how to calculate a sum and an infinite limit. For the sake of simplicity, we will choose the right end point of each subinterval. We can plot this curve and the area beneath it.

[Graphics:HTMLFiles/index_10.gif]

We know that the width of any subinterval must be the total width of the interval divided by the number of subintervals,

In our case this becomes,

We must now choose our point within this interval. This gives us a sequence of subintervals,

Since we are choosing the right endpoint of the subinterval, we can see that

Returning to the summation definition of the definite integral, we have,

We now need to take the limit of this.

There are a couple of interesting points to make. Since this is a summation, the only thing that changes is . is a constant for the purpose of a summation. Since we have the summation of a sum, we note that

So, in our case,

We also know that

So,

$FormBox[RowBox[{ ... ^2, +, RowBox[{1/n, RowBox[{Underoverscript[∑, i = 1, arg3], 10.}]}]}]}]}], TraditionalForm]$

The values of these summations are known

and

So,

$FormBox[RowBox[{ , RowBox[{RowBox[{Underoverscript[∑, i = 1, arg3] f(ξ_i) Δ x_ ... , RowBox[{1/n, RowBox[{Underoverscript[∑, i = 1, arg3], 10.}]}]}]}], ,}]}], TraditionalForm]$

$FormBox[RowBox[{ ... nbsp; , RowBox[{=, RowBox[{8/(6n^3) n (1 + n) (1 + 2 n), +, 10.}]}]}], TraditionalForm]$

$FormBox[RowBox[{ ... nbsp; , RowBox[{=, RowBox[{4/(3n^3) n (1 + n) (1 + 2 n), +, 10.}]}]}], TraditionalForm]$

This is where we apply the limit.

We know that the limit of a sum is the sum of the limits, so,

Now, it is a fact that anything divided by infinty is 0, so,

Thus,

References

[1] Richard A. Silverman, "Modern Calculus and Analytic Geometry," Macmillan Company (1969) (republished in 2002 by Dover Publications).

Created by Mathematica (February 17, 2005)

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