What is the net area between #f(x) = cos^2xsinx # and the x-axis over #x in [0, 3pi ]#?

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Answer 1 · 2016-09-21T16:12:27

#2/3# areal units.

Net area = int f(x) dx, between the limits #0 and 3pi#

#=int cos^2x sin x dx#, between the limits #0 and 3pi#

#=-int cos^2x d(cos x)#, between the x-limits #0 and 3pi#

#=-[(cos x)^3]#, between the limits #0 and 3pi#

#=-((cos(3pi))^3-(cos 0)^3)#

#=-((-1)^3-1)#

#=2/3#

Note that f is periodic with period #2pi# and, Interestingly, the net

periodic area ( here up to #x = 2pi# ) =#2/3-2/3=0#..