What Does an Integral Represent?

Date: 07/09/2004 at 12:58:52
From: Ned
Subject: The meaning of "integral"

My local math support group and I (known among ourselves as The 
Society of Dim Bulbs) have been using Stewart's "Calculus", 5th Ed., 
to "refresh our memories".  On page 373, question 10 says, "Use an 
integral to estimate the sum (from 1 to 10000) of sqrt(i)".  The 
function y(i) is easy enough to graph, and the definite integral easy 
enough to formulate (i.e., 2/3*i^3/2, from 1 to 10000).  The question 
is, what does the integral (the area under the curve) represent?  It 
surely doesn't represent the sum of the square roots as required.

We thought we knew what an integral was, but obviously we don't.

Date: 07/09/2004 at 13:23:25
From: Doctor Ian
Subject: Re: The meaning of "integral"

Hi Ned,

Here's a quick explanation of what an integral is:

  What Is An Integral?
    http://mathforum.org/library/drmath/view/64571.html 

In your case, imagine that we plot some non-negative integers against
their squares:

    |
 16 -               *
    |
 14 -
    |
 12 -
    |
 10 -
    |           *
  8 -   
    |
  6 -
    |
  4 -       *
    |
  2 -
    |   *
 0  *---|---|---|---|--
    0   1   2   3   4

Now let's turn this into a bar chart, where each rectangle has a width
of 1, and a height up to the curve at its horizontal midpoint:

    |
 16 -              -*-
    |             |
 14 -             |
    |             |
 12 -             |
    |             |
 10 -             |
    |          -*-
  8 -         |
    |         |
  6 -         |
    |         |
  4 -      -*-
    |     |
  2 -     |
    |  -*-
 0  *---|---|---|---|--
    0   1   2   3   4

The total area of the bars is just the sum of the squares, isn't it? 
How would that compare to the area under the curve

  y = x^2

over the same domain, i.e., to 

 4.5
  /
  | x^2 dx  ?
  /
 0.5

The integral and the sum won't be exactly the same,

  sum = 1 + 4 + 9 + 16 

      = 30

                      4.5
  integral = [ x^3/3 ] 
                      0.5


           = (1/3)(4.5^3 - 0.5^3]

           = 30 1/3

But the integral _approximates_ the sum.  Do you see why?

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/09/2004 at 15:04:23
From: Ned
Subject: Thank you (The meaning of "integral")

Thanks loads, Ian.  You were a great help.  Our difficulty was, as you
pointed out, in the concept, "approximates".  Taking smaller intervals
in the Riemann sums definitely gets closer to the real value of the
integral.  

- Ned