Answer to Question #349425 in Calculus for Esh

Question #349425

Find the area that us inside r=4 and outside r=4-2sinθ

1
Expert's answer
2022-06-10T10:45:38-0400
4=42sinθ4=4-2\sin \theta

θ1=0,θ2=π\theta_1=0, \theta_2=\pi

A=120π((4)2(42sinθ)2)dθA=\dfrac{1}{2}\displaystyle\int_{0}^{\pi}\bigg((4)^2-(4-2\sin \theta)^2\bigg)d\theta

=120π(16sinθ4sin2θ)dθ=\dfrac{1}{2}\displaystyle\int_{0}^{\pi}\bigg(16\sin \theta-4\sin ^2\theta\bigg)d\theta

=120π(16sinθ2+2cos2θ)dθ=\dfrac{1}{2}\displaystyle\int_{0}^{\pi}\bigg(16\sin \theta-2+2\cos 2\theta\bigg)d\theta

=[8cosθθ+12sin2θ]π0=[-8\cos \theta-\theta+\dfrac{1}{2}\sin 2\theta]\begin{matrix} \pi\\ 0 \end{matrix}

=8π+0(80+0)=8-\pi+0-(-8-0+0)

=16π(units2)=16-\pi ({units}^2)


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