Answer
$A=\frac{\pi}{2}$
Work Step by Step
Given:
$r=sin\theta+cos\theta$
From $0$ to $\pi$
Use the formula for area under the curve in polar coordinates:
$A=\int_o^{\pi}0.5r^2d\theta=\int_o^{\pi}0.5(sin\theta+cos\theta)^2d\theta$
$A=\int_o^{\pi}0.5(sin^2\theta+2cos{\theta}sin\theta+cos^2\theta)d\theta$
Simply with trig identities and evaluate:
$A=0.5\int_o^{\pi}(1+cos2\theta)d\theta=0.5(\theta+0.5cos2\theta)\vert_o^\pi=\frac{\pi}{2}$
