Calculus Integrals of Non Functions
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Sideways Integrals
Integrals of Non Functions

 

 Not all equations qualify as functions.

 To be a function an equation must pass the vertical line test.
 That is, if you can draw a vertical line 
 through more than one point on the graph of an equation.
 It is not a function.
 
 OK, but even if an equation is NOT a function,
 can we find an integral and calculate an area between the function and an axis?
 
 Do you think, for example, we could find the area between the equation:
 

 X = Y2 - 1

 and the Y axis, from Y = -1 to Y = 1?

 

 
 WELL SURE!
 We're hot shot calculus types now,
 we can do ANYTHING!
 (well almost anything)
 
 We work this problem just like we would work the Y = X2-1 problem, 
 ONLY SIDEWAYS!
 
 Since the whole area we want is where X is negative, 
 it's like the ones we did before where Y was negative.
 
 We have to subtract the value where the area is on the negative side of the axis,
 to get a positive answer.
 So we have:

 

 In this problem Y is the independent variable, so dY = 1.
 
 Since dY = 1, we were allowed to tack it on anywhere we needed it
 
 Finding the integral and solving, we get:
 
 
 
 

   copyright 2005 Bruce Kirkpatrick

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