Answer
The area is $4\pi$.
Work Step by Step
Identify the curve:
$r=4\cos\theta$
$r^2=4r\cos\theta$
$x^2+y^2=4x$
$x^2+y^2-4x=0$
$(x-2)^2+y^2=4$
It tells us that the polar equation $r=4\cos\theta$ describes a circle centered at $(2,0)$ with the radius of $2$.
The area is $A=\pi (radius)^2=\pi\cdot 2^2=4\pi$
