Chapter 13: Q. 34 (page 1027) URL copied to clipboard! Now share some education! In Exercises 29–38, find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.The region where the two cardioids r=3−3sinθand r=1+sinθoverlap Short Answer Expert verified As a result, the area of the cardioids' overlapping zone isA=9π2−1538 Step by step solution 01 Given Information Given equations : r=3−3sinθand r=1+sinθ 02 Simplifications The objective of this problem is to find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.Calculate the heart rate.r=3−3sinθandr=1+sinθMark of r=3−3sinθandr=1+sinθThe cardioid values are r=3-3sinθandr=1+sinθMake a solution for herθ1+sinθ=3−3sinθ4sinθ=2sinθ=12⇒,θ=π6,5π6At the pole, tangent3−3sinθ=0sinθ=1⇒θ=π2and 1+sinθ=0 03 Step 3:The area of the cardioids' overlapping zone The area of the overlapping region of the cardioids r=3−3sinθandr=1+sinθcan be represented as A=2∫−π/2π/6 ∫01+sinθ rdrdθ+∫π/6π/2 ∫03−3sinθ rdrdθIntegrate first with regard to r.,A=2∫−π/2π/6 r2201+sinθdθ+∫π/6π/2 r2203−3sinθdθSet the boundariesA=2∫−π/2π/6 (1+sinθ)22dθ+∫π/6π/2 (3−3sinθ)22θA=2∫−π/2π/6 1+2sinθ+sin2θ2dθ+∫π/6π/2 9−6sinθ+9sin2θ2dθA=2∫−π/2π/6 1+2sinθ+12(1−cos2θ)2dθ+∫π/6π/2 9−6sinθ+92(1−cos2θ)2dθIntegrate in relation to θ,A=2θ−2cosθ+12θ−12sin2θ2−π/2π/6+29θ+6cosθ+92θ−12sin2θ2π/6π/2A=232θ−2cosθ−14sin2θ2−π/2π/6+2272θ+6cosθ−94sin2θ2π/6π/2Set the boundaries.,A=2π4−3−38−−3π42+2274π−2712π+33−9382A=9π2−1538As a result, the area of the cardioids' overlapping zone isA=9π2−1538 Unlock Step-by-Step Solutions & Ace Your Exams! Full Textbook Solutions Get detailed explanations and key concepts Unlimited Al creation Al flashcards, explanations, exams and more... Ads-free access To over 500 millions flashcards Money-back guarantee We refund you if you fail your exam. Start your free trial Over 30 million students worldwide already upgrade their learning with Vaia!