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

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Answer 1 · 2016-01-12T08:40:18

Net Area #= 3 pi = 9.42478#

Area #=int_0^pi x*sin x# #dx# +# int_pi^(2pi) x*sin x# #dx# +#int_(2pi)^(3pi) x*sin x# #dx#

#int u# #dv# = #uv - int v# #du#

Let #u=x#,

#dv# = #sin x # #dx#

#v= - cos x#

#du#= # dx#

#int x*sin x# #dx#=#-x* cos x-int (-cos x)# #dx#

#= -x* cos x+int cos x# #dx#

#=-x* cos x+sin x#

Apply the limits # [0, pi], and [pi, 2pi], and [2pi, 3pi]#

the areas are # pi and -3pi and 5pi#

so that the Net Area = #pi+(-3pi)+5pi= 3pi#