Free Area enclosed by polar graphs Practice Problems • Learn To Be

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Which integral represents the area of the shaded $\blue{\text{inner loop}}$ of the limaçon $r=1+2\sin\theta$ ?

Choose 1 answer:

A
$~\displaystyle \int_{\tfrac{7\pi}{6}}^{\tfrac{11\pi}{6}}{\left(1 +2\sin\theta\right)\,d\theta }$
B
$~\displaystyle \int_{\tfrac{7\pi}{6}}^{\tfrac{11\pi}{6}}{\left(\dfrac{1}{2} +2\sin\theta+2\sin^2\theta\right)\,d\theta }$
C
$~\displaystyle \int_{\tfrac{2\pi}{3}}^{\tfrac{4\pi}{3}}{\left(\dfrac{1}{2} +2\sin\theta+2\sin^2\theta\right)\,d\theta }$
D
$~\displaystyle \int_{\tfrac{\pi}{6}}^{\tfrac{5\pi}{6}}{\left(\dfrac{1}{2} +2\sin\theta+2\sin^2\theta\right)\,d\theta }$
E
$~\displaystyle \int_{\tfrac{\pi}{6}}^{\tfrac{5\pi}{6}}{\left(1 +2\sin\theta\right)\,d\theta }$

Area enclosed by polar graphs