Answer
$32.2 $
Work Step by Step
$A=\int ^{\pi /2}_{-\pi /2}\dfrac {1}{2}\left[ r^{2}\right] d\theta =\int ^{\pi /2}_{-\pi /2}\dfrac {1}{2}\left( 4+3\sin \theta \right) ^{2}d\theta =\int ^{\pi /2}_{-\pi /2}\dfrac {1}{2}\left( 16+6\sin \theta +9\sin ^{2}\theta \right) d\theta =\int ^{\pi /2}_{-\pi /2}\left( 8+3\sin \theta +\dfrac {1}{2}\times 9\times \dfrac {1-\cos 2\theta }{2}d\theta \right) =\int ^{\pi /2}_{-\pi /2}\left( \dfrac {41}{4}+3\sin \theta -\dfrac {9}{4}\cos 2\theta \right) d\theta =\left[ \dfrac {41}{4}\theta +3\cos \theta -\dfrac {9}{8}\sin 2\theta \right] ^{\pi /2}_{-\pi /2}=\dfrac {41\pi }{4}\approx 32.2 $
