What Is An Integral?

Date: 10/14/2003 at 02:05:19
From: Antony
Subject: What does an integral sign thingy do?

I'm in eighth grade, and curious about what an integral (S thing) 
does. If it is not too hard for you to explain, I would like to know.

I think it has to do something with graphs and long equations, or 
such.

Date: 10/14/2003 at 10:15:36
From: Doctor Ian
Subject: Re: What does an integral sign thingy do

Hi Antony,

It does have something to do with graphs.  Are you familiar with
functions at all?  For example, if I write

  f(x) = x^2

that's a way of specifing the graph of a particular parabola.  I can
put in values for x, and get values of the function:

   x   f(x) = x^2
  ---  ----------
   1    1
   2    4
   3    9
   4   16

and so on.  If I plot these on a graph, and fill in the points in
between, I have the graph of a parabola:

  |        *                <- (3,f(3))
  |
  |
  |
  |
  |     *                   <- (2,f(2))
  |   
  |
  |  *                      <- (1,f(1))
  +-----------------
     1  2  3

Does this make sense so far?  Now, suppose I want to find the area
between the parabola and the horizontal axis, for some part of the
parabola--for example, between x=1 and x=3:

  |        *                 
  |        #
  |        #
  |       ##
  |      ###
  |     *###                 
  |     ####
  |    #####
  |  *######                 
  +-----------------
     1  2  3

For simple shapes like rectangles and triangles, we have formulas that
we can use to find areas.  But for arbitrary shapes, we use integrals
to find the area.  

The way an integral works is by cutting the area up in to very thin
rectangles, and adding up the areas of the rectangles


  |        *               ~  
  |        6          area =   area of rectangle 1
  |        6                 + area of rectangle 2
  |       56                 + area of rectangle 3
  |      456                 + area of rectangle 4
  |     *456                 + area of rectangle 5
  |     3456                 + area of rectangle 6
  |    23456
  |  *123456                
  +-----------------
     
When we have a finite number of rectangles, we use 'summation' notation:

            6
           __
       ~  \
  area =  /__  area of rectangle i
           i=1


       = area of rectangle 1 + ... + area of rectangle 6

Of course, when we do this, we don't get the exact area!  (We put the
little '~' above the '=' to remind us of that.)  Some of the
rectangles will stick up above the curve, and others won't quite reach
it. So it will just be an approximation to the actual area. 

But as we make the rectangles thinner and thinner, the approximation
gets closer to the true area. For example, if we cut the area into a
million rectangles, we'll get a pretty good idea of the actual area!

           1,000,000
           __
       ~  \
  area =  /__  area of rectangle i
           i=1


And as we imagine that we have an infinite number of rectangles, each
with width zero, the sum 'converges', which means that after a while,
we can figure out what the true value is by looking at the
approximations.  When that happens, we use an integral sign, to
indicate that the summation is continuous rather than discrete:

           x=3
           /
   area =  |   x^2 dx
           /
          x=1

Note that now we just have a regular '=' sign, because this is no
longer an approximation. 

So that's the basic idea.  To see it how it relates to the rest of
calculus, take a look at this:

  A Brief Overview of Calculus
    http://mathforum.org/library/drmath/view/52121.html 

Is this what you wanted to know? Write back if you'd like to talk more
about this, or anything else. 

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

Date: 10/14/2003 at 20:40:11
From: Antony
Subject: Thank you (What does an integral sign thingy do)

Wow! I'm stunned! I really didn't think you would spend so much care
with this! You guys are really an amazing website! 

When I asked my mom what an integral was, she started blabbing on
about something I didn't get a word of.  When you explained it to me,
it's like WHAM, and I know it there and then!  

Thanks. I hope your site never fails.

                                Sincerely,
                                Antony.