Given: r=1+cos(theta)r=1+cos(θ)
Required: Area of cardioid?
Solution Strategy: Polar Coordinate Area Integral
A= int_(theta_1)^(theta_2) 1/2r^2d(theta)A=∫θ2θ112r2d(θ) substitute for rr
A= 1/2int_(theta_1)^(theta_2) ( 1+cos(theta))^2d(theta)A=12∫θ2θ1(1+cos(θ))2d(θ)
=1/2[int (1+2costheta+cos^2theta) d(theta)]=12[∫(1+2cosθ+cos2θ)d(θ)]
= 1/2[theta + 2sintheta+ int cos^2theta d(theta)]=12[θ+2sinθ+∫cos2θd(θ)]
I_3=color(brown)(int cos^2theta d(theta))I3=∫cos2θd(θ) apply reduction formula
I_3=(n-1)/n int cos^(n-2)thetad(theta) +(cos^(n-1)thetasintheta)/nI3=n−1n∫cosn−2θd(θ)+cosn−1θsinθn
I_3=color(brown)(1/2 theta +(costhetasintheta)/2)I3=12θ+cosθsinθ2
Putting it all together:
A=((cosx+4)sinx+3x)/4A=(cosx+4)sinx+3x4
