Questions tagged [order-of-integration]

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

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Center of mass of paraboloid, changing order of integration

I have the following question: Find the center of mass for the following body: A paraboloid $z=a(x^2+y^2)$ between z = 0 and z = b with uniform density $\rho=\rho_0\frac{z}{a}$. I tried to calculate ...
1 vote
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Order of Integration for $\int_{0}^{2}\int_{x^2}^{x}y^2xdydx$

I am trying to evaluate the integral $$I=\int_{0}^{2}\int_{x^2}^{x}y^2xdydx$$ using change of order of integration.Basically the region is $$R=\left\{(x,y):0\leq x\leq 2,x^2 \leq y\leq x\right\}$$ I ...
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Why the order of a series can't be changed

Why do the infinite series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-... \neq 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}...$ ? While when integrating it's possible to ...
• 417
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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 ...
1 vote
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• 225
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Assumptions for stochastic Fubini's theorem for Brownian Motion

I am looking for a basic statement of stochastic Fubini's theorem for Brownian Motion and simple integrands. I have been searching in the internet but I have only been able to find references which ...
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Bounds for a region of integration

I want to determine the bounds for integral $\displaystyle\int\int_R f(x,y)dxdy$, or $\displaystyle\int\int_R f(x,y)dydx$ so that I change the order of the iteration. $R$ is the region lying on the ...
• 2,817
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Changing order of iterated integral

I am given an integral $$\int_0^1dz \int_0^{1-z}dy\int_0^1f(x,y,z)dx$$ and I want to re-iterate it to have integration with respect to $z$ on the inside and integration with respect to $x$ on the ...
• 1,058
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Prove the following expression by changing order of integration

To prove using change of order of integration : $$\int_0^{\pi/2}\int_0^{\pi/2} \sin(x)\arcsin(\sin(x)\sin(y))\,dx\, dy=\frac{\pi^2}{4}-\frac{\pi}{2}$$ Progress so far: Let $\sin x \sin y=\sin z$, ...
• 4,747
In general, I know that for the triple integral $$\iiint f(x,y,z)dxdydz$$ the order of integration cannot be changed arbitrarily. However, if we know that the bounds are constants and that f(x,y,z)=...