Questions tagged [order-of-integration]

For questions concerning the order of integration in multiple integrals, usually involving changing the order of integration.

The arbitrary integral $$\iint_Df(x,y)\,dx\,dy$$ over some region $D$ in the $xy$-plane should, as it stands, be performed by first integrating $f(x,y)$ with respect to $x$ and then by integrating the result with respect to $y$. In some cases, however, it is desirable to change the order of integration, usually to simplify the calculations required.

The order of integration can, of course, be changed in triple and higher-order integrals as well.

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Changing order of integration for $\int_{1}^{2}\int_{2-x}^{\sqrt{2x-x^2}}f(x,y)dydx$

I need to change integration order of $\int_{1}^{2}\int_{2-x}^{\sqrt{2x-x^2}}f(x,y)dydx$ The region is bounded between $1\leq x\leq2$ and $0\leq y\leq1$ The upper limit, $y=\sqrt{2x-x^2}$ in terms of x is: $x=1+\sqrt{1-y^2}$ or…