Answer
$0.248$
Work Step by Step
$A=\int ^{\pi }_{\dfrac {\pi }{2}}\dfrac {1}{2}r^{2}d\theta =\int ^{\pi }_{\dfrac {\pi }{2}}\dfrac {1}{2}\left( e^{-\dfrac {\theta }{4}}\right) ^{2}d\theta =\int ^{\pi }_{\dfrac {\pi }{2}}\dfrac {1}{2}e^{-\dfrac {\theta }{2}}d\theta =-e^{-\dfrac {\theta }{2}}]^{\pi }_{\dfrac {\pi }{2}}=e^{-\dfrac {\pi }{4}}-e^{-\dfrac {\pi }{2}}\approx 0.248$
