Is the jacobian determinant part of the probability density function?

Question

Consider a PDF (Probability Density Function) $f(x,y)$ such that

$\iint_A f(x,y) \text{d}x\text{d}y$ = 1

If I express a similar integral in polar coordinates

$\iint_A f(r,\theta) r \text{d}r\text{d}\theta$ = 1

Then should I interpret $f(r,\theta)$ as a PDF or $f(r,\theta)\cdot r$

Or in other words, should I include the determinant of the Jacobian in my definition of the PDF in polar coordinates (or any coordinate system)?