What is an Integral?

Date: 09/06/2001 at 23:10:04
From: Natasha
Subject: Please explain what an integral is and how it works

Hello, Dr. Math,

I am a grad student who has never had calculus. A few of my courses 
this semester incorporate calculus, including integrals.  I've never 
really understood the concept of Integrals and how exactly they work.  
Could you please explain?

Thanks!

Date: 09/07/2001 at 02:12:45
From: Doctor Jeremiah
Subject: Re: Please explain what an integral is and how it works

Hi Natasha,

They want you to feel challenged in your undergraduate studies so they 
don't tell you this secret (but I can tell you because you are a 
graduate student) The secret: Calculus is easy.

All of calculus was invented just so that people wouldn't ever have to 
graph equations again. Calculus can tell you two things about an 
equation: the slope at any point on the graph, and the area between 
the equation and the x-axis.  Thats it!

When you differentiate an equation you get the slope. When you 
integrate you get the area between equation and the x-axis. These two 
things are opposites (like a square and a square root). For example: 
if you integrate the slope (differential) of an equation, you just get 
the equation back.

How do you calculate the area between the equation and the x-axis?  
Well, you could take two points on the graphed equation and draw a 
trapezoid. That would be accurate if the equation were a straight 
line:

    |                  /
    |                 /
    |                + (x2,y2)
    |               /|
    |              / |
    |             /  |
    |            /   |
    |           /    | 
    |          /     |
    | (x1,y1) +      |
    |        /|      |
    |       / |      |
    +------/--+------+---
          /

If the equation produces a curve there will be an error, because a 
trapezoid needs a straight line along the top. But wait - what if you 
broke it up into a bunch of narrower trapezoids side by side? Then 
there would be less curviness along the top of each trapezoid, and so 
there would be less error in each trapezoid. Then you sum up all the 
areas of all the trapezoids and you get a more accurate total area.  
So mathematically:

(N is the number of trapezoids)

         n = N
         +----
  Area =  \    trapezoid area n
          /
         +----
         n = 1

The sum of all the trapezoids numbered 1 to N.

But there is still some error in the total area because the trapezoids 
still have a bit of curviness at the top, so we make them even 
narrower and use more of them. And our area measurement gets more and 
more accurate. When is it perfect? When we have an infinite number of 
infinitely thin trapezoids.

                         n = N
                         +----
  Area =     limit        \    trapezoid area n
         N -> infinity    /
                         +----
                         n = 1

The sum of all the trapezoids numbered 1 to N as the number N is 
increased to infinity.

But it is way too much work to write all that down, so they came up 
with a simpler method:

          /
          |
  Area =  }  trapezoid area
          |
          /

and that's what an integral looks like. Basically it means that Area 
is the sum of all of the infinite number of infinitely thin trapezoid 
areas.

I hope it seemed simple to you. It really is. If it didn't seem simple 
then I didn't explain it well. If so, please mail me back and give me
a second chance.

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