= Solution
{c}
Uniform area distribution on the unit disk gives
$$
f_{X,Y}(x,y)=\pi^{-1}\mathbf1_{\{x^2+y^2\leq1\}}.
$$
The <Jacobian determinant> has absolute <determinant> $r$, so
$$
f_{R,\Theta}(r,\theta)=\frac r\pi
\mathbf1_{\{0<r<1,\ 0\leq\theta<2\pi\}}.
$$
Integrating out the other coordinate gives
$$
\boxed{f_R(r)=2r\mathbf1_{(0,1)}(r),\qquad
f_\Theta(\theta)=\frac1{2\pi}\mathbf1_{[0,2\pi)}(\theta).}
$$
The <joint probability density function> factors into these two <marginal distribution>, proving <independence> of $R$ and $\Theta$. Finally
$$
\boxed{\mathbb ER=\int_0^1r(2r)dr=\frac23.}
$$
The angle at $R=0$ can be defined arbitrarily, since that event has <probability> zero.
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