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INTEGRATION LIMITS OF A CIRCLE WITH A CIRCULAR VACANCY

I am trying to express the integration limits of the generic region defined by the dotted purple circumferences (of radius Ru-R*=Ru, θ*(Ru)=pi/2, and θ*(z)=pi if z>Ru. The mathematical expressions of the functions θ*(z), r_m(θ), and r_X(θ) are known. Thank you in advance for your help![image at : https://www.freemathhelp.com/forum/attachments/cdf_uav-png.17434/?hash=2068a103ad2738cffb1c8e740f29619d ]