



Up to
now we have been finding the area between an equation graph line



and
an axis, usually
the x axis. 





Now we're
going to find the area between two graph lines. 





Suppose
we have the equations: 








and we
want to find the area in between the places they cross ... 











Well the
first thing is to figure out where the lines cross. 


To do
that, we have to solve the "set" of these two equations. 





The two
points where the equation lines cross, 


are the
two (X,Y) values that make both equations true
"simultaneously." 





There are
a lot of ways to work problems like this, 


but the
way I will do it is here is to substitute the value of Y 


from one
equation into the other equation. 


That ism
substitute Y = X ^{2} into Y = X ^{2} + 2 





Y = X^{2}
+ 2



X^{2}
= X^{2} + 2



2X^{2}
= 2



X^{2}
= 1



X
= ±1






So the
area we want goes from X = 1 to X = 1. 





The next
thing we need to know is which equation is on the top. 


Just
looking at the picture, we can see that it's Y = X^{2} + 2. 


But if we
couldn't tell, we would just pick some point in the interval 


(like
maybe X = 0) and see
which equation had the greater value. 





X = 0 
Y
= 

X^{2} + 2 
Y
= 
X^{2} 
Y
= 

0^{2} + 2 
Y
= 
0^{2} 
Y
= 
2 
Y
= 
0 
The
Winner! 








OK, so Y
= X^{2} + 2 is on top. 


NOW
WHAT???? 





TO
FIND THE AREA BETWEEN TWO LINES, USE THIS FORMULA: 








So here
we have: 








If you
come up with an area of zero for one of the integrals, 


it
probably means you lost a minus sign along the way. 





Example: 





If we
have those same two equations from the last problem 


and we
want the area between them from X = 2 to X = 2, 


we need
to calculate three pieces of area ... 








This is
like when you had one equation 


that had
parts above the axis and parts below the axis. 





So we
divide the area into sections at the places the lines cross. 


And
always subtract the equation that is on the bottom in that section. 


To find
which one that is, 


just
check any X value in the section. 





In this
case, we would have: 








When we
find the area between two equations, 


we don't
need to know if they are above or below the X axis. 


It
doesn't matter. 





OPTIONAL
IDEA: 


Actually,
we've ALWAYS had two equations. 


When we
thought we only had one, 


the other
one was F(x) = 0. 





When our
one and only equation was above the X axis, 


we were
actually subtracting zero. 


(and
subtracting zero doesn't change things a lot) 











When our
one and only equation is BELOW the X axis, we have: 











Again the
integral of zero doesn't matter, and this time we are left with: 











Which is
what it always was, but now we know why. 





copyright 2005 Bruce Kirkpatrick 
