



What if
we wanted to find the area between the X axis and the function ...






F(x) =
sinX + cosX






and we
want all of the area from X = 0 to X = p
radians. 


(for
those not fluent in radianese, that's from 0 to 180 degrees) 





The first
thing we need to do is figure out if the function is always
positive, 


or always
negative, or crosses the axis, or what. 





When X =
0, cosX = 1 and sinX = 0. 


So the
total function value is 1. 


That
means our function starts out ABOVE the X axis 





Now lets
see if the function ever touches the X axis? 


That is,
is there a point where: 


sinX
+ cosX = 0






Let's
find out ... 


sinX
+ cosX = 0



sinX
=  cosX






In
quadrant II (90 to 180), sine is positive and cosine is negative. 


The sine
and cosine are the same (except maybe for sign on the number) 


at 45,
135, 215, 305 (45 + 90 x n). 


The
version of that in quadrant II is 135, also known as 3p/4
radians. 





So we
have two sections. One from 0 to 3p/4
radians 


and one
from 3p/4
radians to p
radians. 





Since the
graph line is above the X axis at X = 0, 


we know
the first section is ok. That is, that area will turn out positive. 





How do we
tell if the second section will be positive or negative? 





Take any
point in that section (excluding 3p/4
), 


and test
to see if the function has a positive or negative value. 





So let's
test the function at p. 


F(x)
= sinX + cosX



F(p)
= sinp
+ cosp



F(p)
= 0  1 


F(p)
=  1






So the
section of the curve to the right of 3p/4
is below the X axis. 


The graph
looks something like this: 











So the
problem works out this way ... 











That
didn't hurt all THAT much did it? 


Well at
least it's over ... 





copyright 2005 Bruce Kirkpatrick 
