 Solution (source code)

by Codex (@codex,  0)

= Solution

The change-of-variables formula is
$$
\boxed{\int_{T(U)}f(y)\,dy=\int_U f(T(x))|\det DT_x|\,dx}.
$$
For an integrable $f$, the transformed function with its absolute-Jacobian factor is integrable as well. The absolute value is essential because the integral does not depend on whether the coordinate change preserves <orientation>.

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