Answer
$2+\dfrac{\pi}{4}$
Work Step by Step
Given: $r=1+\cos 2 \theta$
The area is given as follows:
$A=(4) \int_{0}^{\pi/4} (\dfrac{1}{2}) [1+\cos 2 \theta]^2 d\theta$
Then, $(2)[\sin 2 \theta +(1/2) \theta +\dfrac{\sin 4 \theta}{8}]_{0}^{\pi/4}=2(1+\dfrac{\pi}{8})$
After solving, we get $A=2+\dfrac{\pi}{4}$
