Answer
$\dfrac {\pi ^{2}}{8}+1\approx 2.23$
Work Step by Step
$dA=ydx=\left( x+\cos x\right) dx\Rightarrow $
$$\int dA=A=\int ^{\frac {\pi }{2}}_{0}\left( x+\cos x\right) dx=\left( \dfrac {x^{2}}{2}+\sin x\right) ]^\frac {\pi }{2}_0=\dfrac {\pi ^{2}}{8}+1-0\approx 2.23$$
