What is the net area between #f(x)=sinxcosx# in #x in[0,2pi] # and the x-axis?

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Answer 1 · 2015-12-05T16:05:21

#2# units

We may rewrite this function by making use of the double angle trig identities :

#f(x)=sinxcosx=1/2sin(2x)#

We now draw the graph of this function to help decide limits of integration

graph{1/2sin(2x) [-4.933, 4.934, -2.466, 2.467]}

Since this graph is symmetric with period #pi#, the graph makes 2 full cycles in the interval #[0, 2pi]#

The net are area bounded between the graph and the x-axis is hence 4 times the area bounded in #[0,pi/2]#.

ie. #Area = 4 int_0^(pi/2)1/2sin(2x)dx#

#=-4*1/2*1/2[cos2x]_0^(pi/2)#

#=-(-1-1)#

#=2#