How do you find the area of the region shared by the cardioid #r=2(1+cos(theta)) #and the circle r=2?

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Answer 1 · 2015-09-26T01:18:33

#5pi-8#

Let's find the points of intersection:

#2(1+costheta)=2#
#1+costheta=1#
#costheta=0#
#theta=pi/2 vv theta=(3pi)/2#

enter image source here

We can see from the graph that (considering the half of the area due to the symmetry and doubling the integrals):

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

#A_1=4theta|_(0)^(pi/2)=2pi#

#A_2=4int_(pi/2)^(pi)(1+2costheta+cos^2theta)d theta#

#A_2=4(theta+2sintheta+int_(pi/2)^(pi)(1+cos2theta)/2 d theta)#

#A_2=4(theta+2sintheta+theta/2+1/4sin2theta)|_(pi/2)^pi#

#A_2=4((3pi)/2)-4((3pi)/4+2)=6pi-3pi-8=3pi-8#

#A=2pi+3pi-8=5pi-8#