



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