Question

Help please!
Area shared by the circle r = 5 and the cardioid r= 5(1+sin(theta)

Answer 1

first off the region from theta=0 to theta=pi will be shared (since 5+5sin(theta)>=5 in this region). this area is just pi*5^2/2. now, for the region from theta=pi to theta=2pi, we need to take an integral. 5+5sin(theta)<=5 in this region. Thus, the area integral will be integrate(r^2/2 dtheta)=1/2*integrate(router^2-rinner^2 dtheta)=
1/2*integrate(5^2-(5+sin(theta))^2 dtheta) as theta goes from pi to 2pi.

Answer 2

the total shared area is 25pi/2+the integral

Answer 3

Thank you!