Answer
$ \frac{3\pi}{4} \approx2.36$
Work Step by Step
$A=\int ^{\pi }_{0}\dfrac {1}{2}\left[ r2\right] d\theta =\int ^{\pi }_{0}\dfrac {1}{2}\left[ 1+\cos \theta \right] ^{2}d\theta =\dfrac {1}{2}\int ^{\pi }_{0}\left[ 1+\cos ^{2}\theta +2\cos \theta \right] d\theta =\dfrac {1}{2}\left( \theta ]^{\pi }_{0}+2\sin \theta ]^{\pi }_{0}+\int ^{\pi }_{0}\dfrac {1+\cos 2\theta }{2}d\theta \right) =\dfrac {1}{2}\left( \dfrac {3\theta }{2}]^{\pi }_{0}+2\sin \theta ]^{\pi }_{0}+\dfrac {1}{4}\sin 2\theta ]^{\pi }_{0}\right) = \frac{3\pi}{4} \approx2.36$
