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I want to find the following: $$ \int_{0}^{2}\int_{y/2}^{1}y{\rm e}^{-x^3} \,{\rm d}x\,{\rm d}y $$ using change of variables. I've solved $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} {\rm e}^{-x^2}\,{\rm d}x\,{\rm d}y $$ by changing to polar coordinates, but it doesn't look like that will be as helpful in this case.

Any ideas for a transformation I should use $?$.

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1 Answer 1

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You might sketch the region of integration and you can see quite clear how the change from $dxdy$ to $dydx$ affects the integral. $I = \displaystyle \int_{0}^1 \displaystyle \int_{0}^{2x} ye^{-x^3}dydx$. From this you can finish it.

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