What is the formal definition of " Express the double integral in polar coordinate "?

Question

I was reading Tom Apostol calculus volume 2 and came across, the change of variable for double integral (page 392), the exercise followed by the sections (section 11.28), the very first question of nearly :-

"Express the double integral in polar coordinates"

Now my question is What is the formal definition of the above statement ?

Answer 1

You presumably have an integral $\int \int f(x,y)\;dx\; dy$ over some region where $x$ and $y$ are Cartesian coordinates. You are expected to write an equivalent integral $\int \int g(r,\theta) r\; dr\; d\theta$ over the same region with $r, \theta$ being polar coordinates.

Answer 2

The statement simply means that you express the variables $x$ and $y$ in terms of $r$ and $\theta$. Any point in the plane can be uniquely expressed by a vector with length $r$ from the origin forming an angle $\theta$ with the positive x-axis in the counterclockwise direction. I have not opened the book you are referencing, but say you are solving the double integral of $1$ over the disk $x^2 + y^2 \leq 1$. This integral can be transformed in terms of $r$ and $\theta$ as follows: $$\int\int_D dxdy = \int_0^{2\pi} \int_0^1 drd\theta$$ where $D$ is the domain of integration (here the unit disk).