cosθ= 1/2

θ= π/3

So the intersection of the two curves is (3/2, π/3).

Sketch is easy to understand. The common part enclosed by the curve ρ=3cosθ (that is, a circle with a radius of 3/2 centered on the point (3/2,0)) and the curve ρ= 1+cosθ is a parabola (actually, it is also a quadratic curve and a quartic equation) with the X axis as the symmetry axis. Because it is symmetrical up and down, only y>0 times 2 is calculated.

S=2[∫(0，π/3) 1/2*( 1+cosθ)？ dθ+∫(π/3，π/2) 1/2*(3cosθ)？ dθ]

=∫(0，π/3)( 1+2cosθ+cos？ θ)dθ+∫(π/3，π/2)9cos？ θdθ

=∫(0，π/3)[ 1+2 cosθ+(cos 2θ+ 1)/2]dθ+∫(π/3，π/2)9(cos2θ+ 1)/2dθ

=[3θ/2+2 sinθ+ 1/4 * sin 2θ]|(0，π/3)+[9θ/2+9/4*sin2θ]|(π/3，π/2)

=(π/2+9√3/8)+(3π/4-9√3/8)

=5π/4