Answer to Question #204677 in Calculus for hayrunisa

Question #204677

Calculate the area of the region inside the cardioid r = a(1 + sin Ɵ) outside the r = a sinƟ circle and above the polar ray. Show by drawing.


1
Expert's answer
2021-06-09T13:35:35-0400
r=a(1+sinθ)r=a(1+\sin \theta)

r=asinθr=a\sin \theta


0θπ0\leq \theta \leq \pi




The area of the circle


A1=120π(asinθ)2dθ=a240π(1cos(2θ))dθA_1=\dfrac{1}{2}\displaystyle\int_{0}^{\pi}(a\sin \theta)^2d\theta=\dfrac{a^2}{4}\displaystyle\int_{0}^{\pi}(1-\cos(2 \theta))d\theta

=a24[θ12sin(2θ)]π0=πa24(units2)=\dfrac{a^2}{4}[\theta-\dfrac{1}{2}\sin(2\theta)]\begin{matrix} \pi \\ 0 \end{matrix}=\dfrac{\pi a^2}{4}(units^2)

The area of the part of the cardioid


A2=120π(a(1+sinθ))2dθA_2=\dfrac{1}{2}\displaystyle\int_{0}^{\pi}(a(1+\sin \theta))^2d\theta

=a220π(1+2sinθ+1212cos(2θ))dθ=\dfrac{a^2}{2}\displaystyle\int_{0}^{\pi}(1+2\sin \theta+\dfrac{1}{2}-\dfrac{1}{2}\cos(2 \theta))d\theta

=a22[32θ2cosθ14sin(2θ)]π0=\dfrac{a^2}{2}[\dfrac{3}{2}\theta-2\cos\theta-\dfrac{1}{4}\sin(2\theta)]\begin{matrix} \pi \\ 0 \end{matrix}

=a2(3π4+2)(units2)=a^2(\dfrac{3\pi}{4}+2)(units^2)


Area=A2A1=3πa24+2a2πa24Area=A_2-A_1=\dfrac{3\pi a^2}{4}+2a^2-\dfrac{\pi a^2}{4}

=(π2+2)a2 (units2)=(\dfrac{\pi }{2}+2)a^2\ (units^2)



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