# When to use the Functional Determinant in Polar Coordinate Transformation

I am currently learning about polar coordinate transformation, especially for integrating over certain regions. Let's say we have to calculate

$\int_{n}{xy \; dx dy}$

Then I think the correct transformation would be (note the functional determinant $r$)

$\int_{m}{r cos \phi \cdot r sin \phi \cdot r \; d\phi dr}$

First question: Is that correct?

My second question is: I have some examples a friend of mine once wrote for an exam. The task was for the following region

$B = \{ (x,y) \in R : 0 \leq x^2 + y^2 \leq 1, x \leq y \}$

to calculate the integral

$\int_{B}{x^2+y^2 \; dx dy}$

And he began with

$\int_{B}{x^2+y^2 \; dx dy} = \int_{0}^{\frac{5\pi}{4}} \int_{0}^{1} r^2 sin^2 \phi + r^2 cos^2 \phi \; d\phi dr$

Second question: Where is the functional determinant here? Is it missing or am I wrong assuming there should belong one?

I'd be happy if anyone could help me with some insight on this. Thanks!

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Yes, the determinant is $r$. Your friend made a mistake, there should be an $r$ there too (and the lower limit for $\phi$ should be $\pi/4$).