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

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Answer 1 · 2017-06-07T16:35:00

#A=1/3sin^3 3pi#

The question is asking us to find #int_0^(3pi)cosxsin^2x# #dx#

#intcosxsin^2x# #dx#

Let #u=sinx#

#du = cosx# #dx#

#dx= secx# #du#

#intcosxsin^2x# #dx= int u^2cosx xx secx# #du#
#= intu^2# #du=1/3u^3 + "c" = 1/3sin^3x +"c"#

#int_0^(3pi)cosxsin^2x# #dx= [1/3sin^3x]_0^(3pi) #
#= 1/3sin^3 3pi -0 = 1/3sin^3 3pi#