Chapter 10 - Parametric Equations and Polar Coordinates - 10.4 Areas and Lengths in Polar Coordinates - 10.4 Exercises - Page 712: 1

0.248

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