Why is it impossible to have different integration results in cartesian and polar coordinates?

Question

Let $A$ be a surface in $\mathbb{R}^2$.

Let $f:A \rightarrow \mathbb{R}$ be a Riemann integrable function on $A$.

We have to find the multiple integral of function $f:A \rightarrow \mathbb{R}$.

By definition of Riemann Integral:

• In Cartesian coordinates, $\iint_A f(x,y)\ dA=\text{lub} \{ L(P,f) \}$ where $P$ is any rectangular partition of $A$.

• In polar coordinates, $\iint_A f(r,\theta)\ dA=\text{lub} \{ L(P',f) \}$ where $P'$ is any circular sector partition of $A$.

I know $\text{lub} \{ L(P,f) \}$ and $\text{lub} \{ L(P',f) \}$ are equal? But how can we prove this?