Transform $\int^a_b\int^c_df(x,y)dxdy$ into $\int^k_l\int^m_nf(r,\theta)drd\theta $. 
Is there any general way to transform $\int^a_b\int^c_df(x,y)dxdy$ into $\int^k_l\int^m_nf(r,\theta)drd\theta $? If yes, what is it?

I'm a new double integral learner and just have the idea of polar coordinate. Can anyone help me with this Question? Thank you.

No, if you intend to have constant $a, c, b, d$ and $k, l, m, n$, this is not possible, because in the $xy$ domain you have a rectangle, and its edges have equations, say,
$$
x = A\\
y = B
$$
where $A,B$ are constant, and these become in the $r\theta$ domain
$$
r=\frac{A}{\cos\theta}\\
r=\frac{B}{\sin\theta}
$$
which cannot be constant, if we exclude the simple case of $A$ or $B$ zero.
