Is the below true?
$$\int_{y_1}^{y_2}\int_{x_1}^{x_2}f(x,y)dxdy \overset{?}{=} \int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)dydx$$
2026-04-19 03:58:10.1776571090
Does changing the order of definite integration change the result?
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$$\int_{y_1}^{y_2}\int_{x_1}^{x_2}f(x,y)dxdy \overset{?}{=} \int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)dydx$$ This in general is false, consider in fact $x_1=y_1=0$ and $x_2=y_2=1$ and define $f(x,y)$ as follow: $$f(x,y)=\begin{cases} y^{-2}\quad \quad0<x<y<1 \\ -x^{-2} \quad \,0<y<x<1 \\ 0 \quad\quad \quad\text{otherwise} \end{cases} $$