How do you find the area of the region enclosed by the cardioid #r=2+2cos(theta)#?

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Answer 1 · 2015-09-20T16:30:41

#6pi#

#A=intint_(D)rdrd theta#

#A=int_(0)^(2pi)d thetaint_(0)^(2+2costheta)rdr=1/2int_(0)^(2pi)d theta r^2|_(0)^(2+2costheta)#

#A=1/2int_(0)^(2pi)(2+2costheta)^2d theta#

#A=1/2int_(0)^(2pi)(4+8costheta+4cos^2 theta) d theta#

#cos^2 theta = (1+cos 2theta)/2#

#A=1/2int_(0)^(2pi)(4+8costheta+2+2cos2theta) d theta#

#A=1/2(4theta+8sintheta+2theta+sin2theta) |_0^(2pi)#

#A=6pi#