I'm trying to prove this integral $$ \int_0^a \int_0^\sqrt{a^2-x^2} f(x,y) \, \mathrm{d}y \, \mathrm{d}x$$ is the same as $$\int_0^{2\pi} \int_0^a r f(r,\theta) \, \mathrm{d}r \, \mathrm{d}\theta$$ I believe this is not hard, maybe just impose a Jacobian or something? Basically it's just integrating a function in a circle area. I've been trying but failed. Could anyone please help me out? Thanks!!!
2026-03-30 04:23:25.1774844605
A polar integration question
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Do you know what a Jacobian is? In that case, it is not hard. The Jacobian for polar coordinates is $r$, so you simply multiply by that. Note that the first integral is only over one quarter of the circle, so the second should have $0\le\theta\le{\pi\over 2}$.