Answer
There is a factor $r$ in the integral that makes the difference. This implies that the area in polar coordinates depends on its distance from the origin.
Work Step by Step
In ordinary rectangle, the double integral of $f\left( {x,y} \right)$ is given by
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} f\left( {x,y} \right){\rm{d}}A$,
where the infinitesimal area is ${\rm{d}}A = {\rm{d}}x{\rm{d}}y$.
The Change of Variables Formula for polar coordinates is
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} f\left( {x,y} \right){\rm{d}}x{\rm{d}}y = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{r = {r_1}}^{{r_2}} f\left( {r\cos \theta ,r\sin \theta } \right)r{\rm{d}}r{\rm{d}}\theta $,
where in polar coordinates, ${\rm{d}}A = r{\rm{d}}r{\rm{d}}\theta $.
Notice that there is a factor $r$ in the integral that makes the difference. This implies that the area in polar coordinates depends on $r$, its distance from the origin.
