I have the following past exam question.

Write down the Cartesian coordinates x and y in terms of the plane polar coordinates r and $\displaystyle \theta$

Which is easy.

Next, evaluate the jacobian determiant

By considering the matrix equation relating the vectors

$\displaystyle \begin{pmatrix}{dx}\\{dy}\end{pmatrix}$

and

$\displaystyle \begin{pmatrix}{dr}\\{d\theta}\end{pmatrix}$

use this result to obtain an expression for the area element dxdy in plane polar coordinates.

b)Work out the numerical value of the same area integral

$\displaystyle \int(r^2+1)dA$

over the interior of the circle of radius 2 centred at the origin.