Answer
True
Work Step by Step
By Fubini's Theorem, we have $\int^{b}_{a}\int^{d}_{c}f(x,y)dydx=\int^{d}_{c}\int^{b}_{a}f(x,y)dxdy$
so $\int^{1}_{0}\int^{2}_{-2}xe^{y}dydx=\int^{2}_{-2}\int^{1}_{0}xe^{y}dxdy$ is correct.
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