Calculus Area Between Two Lines
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From Here to There
Area Between Two Lines

 

 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 = -X2 + 2 

 X2 = -X2 + 2 

 2X2 = 2 

 X2 = 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 = -X2 + 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 =   - X2 + 2 Y =   X2
Y =   - 02 + 2 Y =   02
Y =   2 Y =   0

The Winner!

 
 OK, so Y = -X2 + 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

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