



Say we
want to find the area between the line of some equation and the X
axis.



Like
maybe, between F(X) = X
^{2} and the X axis from X = 0 to X =
1 





OK,
This is not some silly phoney baloney exercise 


to practice moving
symbols around, 


this
is very valuable calculus stuff so pay attention here. 





So
we have something like: 








This
is a weird shape. 


We
don't have a formula from geometry for a shape like this. 


We
could get a good guess at the actual area of the shape 


by
dividing it up into a bunch of little rectangles and adding those
areas. 


And
calculating the area of a rectangle is easy. 








When
we do this, 


there
are little spaces above the rectangles 


that should be included in
the area but aren't. 


So
we can see right away that our answer won't be exactly right. 


And
it will be too low. 


We
can also see that if we have more thinner rectangles 


less
area at the top of the rectangles will get left out, 


the
closer we will get to the right answer. 








If
we divide the area into 3 rectangles, each one is 1/3 wide. 


If
we divide the area into 50 rectangles, each one is 1/50 wide. 


So
in general, 


If
we divide the area into "n" rectangles, each one is 1/n
wide. 





So
we have: 








BUT
HOW TALL ARE ALL OF THESE RECTANGLES??? 





If
we have 50 rectangles in our 1 unit length of X, 


the
first one goes from X = 0 to X = 1/50, 


the
second one goes from X = 1/50 to X = 2/50 


the
third one goes from X = 2/50 to X = 3/50 


and
so on ... 








These
numbers are values of X, we can plug them into our equation! 





F(x) = X^{2}






The
function value at each X is the height of that rectangle. 


One
More Time ... 


The
function value at each X is the height of that rectangle. 





We
can actually use the X value on the left side of the rectangle 


or
the X value on the right side of the rectangle. 





If
we use the left side numbers: 0/50, 1/50, 2/50, 3/50, etc. 


We get
these rectangles ... 








If
we use the right side of the rectangle height values, 


1/50
for the first, 2/50 for the second, and so on. 


We
get: 








If
we use the first way (left side X values), the area we get is a bit
too small. 


If
we use the second way (right side X values), the area we get is a
bit too big. 


BUT,
if we use lots of tiny rectangles, both answers will be pretty
close. 





If
we want the area between the curve and the X axis from X = 0 to X =
1 


Think
of it as the sum of the area of lots of little rectangles. 


Let's
say ... 50 of them (you could use a million if you want) 


Each
one is 1/50th of a unit wide. 


To
calculate the height of each rectangle, 


use the
X values from one side or the other of each rectangle. 


Since
our function is F(x) = X ^{2}, we square each of the X values
to get the height.






The
area of each rectangle is the width 1/50 times the height X^{2} 


So
using the left side of each rectangle X values, we get: 











Did
you notice that we stopped at 49/50 


because we were using the left
side X values??? 





We
can use the
S
notation to write this, 


we
would have: 








Once
we pull all of the constants away, we will be evaluating X^{2}
from X=0 to X=49 


The
problem is, that the formulas we have from the last page 


only work
when we start at X=1. 


We
have two choices. 


We
can use right side of the rectangle X values so we go from X=1 to
X=50 


OR 


We
can show you a trick to deal with starting from zero 


Guess
which way we're going ... 





Here's
the trick 


Since
we have X/50 and are starting at 0, we can use (X1)/50 and start at
1 








Moving
constants left: 





Separating
the numerator and denominator: 





Moving
more constants left: 





Multiply
out the right side: 





Separating
terms: 





Now
we can use those summation formulas 


from
the last page: 





n
= 50, so: 





Get
out your calculator ... 


Area » 
1 
(42,925
 2,550 + 50) 

250,000 






Area
»
0.3234






If
we do this same process using the right side of the rectangle X
values: 











Since
we used right side X values, we went to 50/50. 


The
S
notation for this one is: 





Which
can be written as: 





Moving
all constants left: 





So
the answer is: 








As
we figured, this answer is a bit bigger. 


The
real answer must be somewhere between these two numbers. 





Most
people would probably say: 


"Hey
just average the two and that's good enough for me!" 





But
of course, math people AREN'T most people. 


So
they plodded along looking for really weird equations 


that give us
even better answers! 


Aren't
we lucky! 





Our
general form of the F(X) = X
^{2} area equation is: 











In
the equation, "n" is the number of pieces we cut our
horizontal distance into. 


And
n is the counting sequence from 1 to whatever we choose n to be. 





The
greater the number of rectangles we use 


the closer the area we get will be to
the real area. 





The
greater number of rectangles we use the smaller each rectangle will be. 


As
the number of rectangles gets bigger the width of X (call it DX)
gets smaller. 


If
DX
gets as absolutely small as it could ever get ... 





OH
NO!



I
HEAR A LIMIT PROBLEM APPROACHING!






That's
right. As the number of rectangles approaches infinity, 


the
area gets as close as it can possibly get to the actual area. 


We
can even say ... 








So
lets solve this. 


Move
out the constant and divide up the other stuff ... 








Move
the constant to the left... 








Use
the rules and evaluate the summation ... 













So
the real answer is between the two rectangle answers. 


We
KNEW it would be. 





Ya
know, this worked and all, 


but
as the equations get more complicated this is going to be a bear 


I
wish there were some shortcuts to this limits process. 





Is
this like deja vu or what? 





copyright 2005 Bruce Kirkpatrick 
