



How do
we calculate the area between a curve and the X axis



from one X
value to another?






Say
we want the area between the line F(X) = X
^{2} and the X 


axis from X = 0 to X = 1. 


We
set up an integral notation and put the smaller number at the
bottom 


and
the bigger number at the top. 





Find
the integral of X ^{2}dX, 


and
place it in square brackets. 





The
first step is to substitute the top number in for X 











The
last step is to substitute the lower number into the term 


and
subtract that from what we already have. 











And
we're back to the same value, 1/3. 





OK,
now lets find the area between the F(X) = X
^{2} curve and
the X axis 


from
X = 5 to X = 20 ... 




































One
nice thing about F(X) = X
^{2}, is that it never crosses the
X axis. 











What
happens if we have an equation that does cross the X axis? 





What
if we have an equation like: 


F(X)
= X^{3}






And
we want to find the area between the curve 


and
the X axis, from X = 2 to X = 2? 








If
we just plug away, we get: 




































OOPS! 


I
can see we have some area here, 


how
come we got zero for an answer? 





When
left on their own, 


the
equations will see any area below the axis as NEGATIVE area, and
subtract it. 





If
there is more area below the axis, 


the
whole answer will be negative. 





OK,
How come the area below the X axis is negative area? 





Let's
talk about how the equations "build" the area. 





Say
we have the equation: 


F(X) = X^{2}






The
graph looks like this: 








The
area under the graph from X = 0 to X = 3 


looks
like this: 





The
graph of the equation 


F(X) = X^{2}
+ 5






Looks
like this: 








The
thing is, the area between the curve and the X axis is made up of
two pieces. 


One
is the same odd shape area we had with F(X) = X
^{2}. 


The
other is a rectangle 5 units high and 3 units wide. 








When
we find the integral, 


we even wind up with these two area pieces as
separate terms. 











The
graph of an equation like F(X) = X
^{2}  5, looks like this: 











The
area between this equation line and the X axis, 


from
X = 0 to X = 3 is 





The
area is: 





The
way the equation builds this is: 








So
how do we get all of the area parts to give us positive numbers?, 





OK,
say we have this equation ... 


F(X)
= X^{2}  1






and
we want to find the total of all the areas between this curve 


and
the X axis from X = 2 to X = +2 








Say
we want the total area, including all of these three pieces. 


That
is, we want to count them all as positive areas. 





OK,
so we know that if we just run the numbers, 


the
middle piece will come out as negative. 


To
do this right, we need to find out the X values 


where
that middle piece starts and ends. 


Those
are the places it crosses the X axis. 


AND
AT THOSE PLACES THE FUNCTION EQUALS ZERO! 





So
set the function equal to zero and solve for X ... 





F(X)
= 
X^{2}
 1 
0
= 
X^{2}
 1 
X^{2}
= 
1 
X
= 
±
1 






So we
have three sections, X = 2 to X = 1, X = 1 to X = 1, And X = 1 to
X = 2. 


Two of
these are above the X axis and one is below it.



To make
the area of the one below the X axis work out positive,



we
need to subtract it. 


That
makes the whole thing look like this: 











There
are a whole lot of places where you could make a mistake 


on
something like this, 


SO
BE CAREFUL! 


Losing
track of a sign is a VERY COMMON MISTAKE 





The
fact that all of the numbers in that last step were either plus or
minus 2/3 , 


probably
has some cosmic significance. 


But
we really aren't worried about that right now. 





copyright 2005 Bruce Kirkpatrick 
