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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Curious interchange of the order of summation
Usually interchanging the order of summation requires $\sum_n\sum_m|a_{n,m}|<\infty$. Unfortunately, I don't have this condition on my hands. Only conditions I have are:
$$\sum_n\left|\sum_m a_{n,m}...
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Switching the order of a triple integral and differing results, a Multivariable Calculus exercise
Here is a triple integral a student asked me, but which I can't seem to get right. I have tried to recalculate the integrals several times, but to no avail. If you're taking a Multivariable Calculus ...
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Change the order of integration: $\int_0^1 \int_0^1 \int_{x^2}^1 12 xz \exp(z y^2) \mathrm dy \mathrm dx \mathrm dz$
Consider the following integral:
\begin{equation}
\int_0^1 \int_0^1 \int_{x^2}^1 12 xz \exp(z y^2) \ \mathrm dy \ \mathrm dx \ \mathrm dz.
\end{equation}
It has been said that the integral cannot be ...
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How do I change the order of integration for this integral?
The question is related to Buffon's needle experiment: I was wondering how to change the order of integration. The integral is shown below. Apologies if I have left out lots of stuff, I'm quite new to ...
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Question on changing order of integration
Let $f(x)$ be a positive continuous pdf with support over $[a,b]$ and let $F(x)$ be its corresponding cdf. Let $p: [a,b] \to [0,1]$. I am trying to figure out the following chain of equalities:
$$ \...
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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 ...
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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 ...
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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 ...
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Does $Y \in b(X) \iff X \in c(Y)$ implies $E[g(X,Y)\mid Y \in b(X)] = E[g(X,Y) \mid X \in c(Y)]$?
Suppose I have two random variables $X$ and $Y$ that are i.i.d. according to a prob. measure $\mu$ whose support is $A$. Take a function $g:A\times A \to \mathbb{R}$ such that $E[g(X,Y)\mid Y \in b(X)]...
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Conditions to change order of integration
I was reading a proof on Fourier's Integral Theorem, I won't get too much into that because the reason for the question doesn't really have a lot to do with it so I will just mention the steps which ...
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Changing order of integration using chart technique error?
Consider: $$ \int_0^3 \int_4^{\sqrt{25-z^2}} \int_{-\sqrt{25-y^2-z^2}}^{\sqrt{25-y^2-z^2}} dxdydz$$
i. Clearly sketch the graph of the solid whose volume this triple integral determines.
ii. Present ...
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Why should we take one degree less numerator in Partial Fraction Integration?
In the method of Integration using partial fraction method, the numerator is always kept to have 1 degree less than that of the denominator.
Say, numerator would be Ax+B for a second degree ...
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integral using change of order of $\int _0^9\:\int _{x^{1/2}}^3\cos{(y^3-3y)}\,dy\,dx$?
I was refreshing multiple integral, and I came across with this question:
$\int _0^9\int _{x^{1/2}}^3\cos{(y^3-3y)}\,dy\,dx$
I used change of order since $\cos(y^3-3y)$ cannot be integrated at the ...
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A triple integration that need change of boundary
Evaluate Integral: (Without calculator, Just by hand)
$$\int_0^1\int_0^{1-x}\int_y^1\frac{\sin(\pi z)}{z(z-2)}\,dz\,dy\,dx$$
The answer of the problem uses a visualization of the boundary and ...
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Order of integration in triple integral
Is there any hard and fast rule for what order you integrate for triple integrals. I know of Fubini's theorem but surely this doesn't cover all cases of triple integrals.
Say for example I have,
$$\...
user571032
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Evaluating an improper integral using double integrals [duplicate]
working on a problem to evaluate
$\int_0^\infty \frac {e^{-x} - e^{-ax}} {x} dx$
the instructions say to first evaluate
$\int_1^a e^{-xy} dy$
which comes out to the integrand of the original ...
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votes
2
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Differential forms and order of integration
I don't understand how $$
\int_{a_2}^{b_2} \int_{a_1}^{b_1} f(t_1,t_2) dt_1 dt_2 =
\int_{a_1}^{b_1} \int_{a_2}^{b_2} f(t_1,t_2) dt_2 dt_1
$$ can agree with the fact that $dt_1 \wedge dt_2 = -...
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General procedure to switch order of the integrals
I am having problems in understanding how the limits of the integral change when we switch the order of integration. In particular, let $f,g : \mathbb{R} \rightarrow \mathbb{R}$ be integrable ...
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Different results after changing the order of integration with constant limits (Failure of Fubini's theorem)
I have the following question
$I_{1}=\int _{0}^{1}\int _{0}^{1}\ \frac{(x-y)}{(x+y)^{3}}\ dy\,dx$
Evaulating the above I get $I_{1}=0.5$
Now if I switch the order of integration
$I_{2}=\int _{0}^{...
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Changing order of integration restricted by square root and circle
This is not a homework (it's an exam preparation exercise set). I have an issue with changing the order of integration of the following integral:
$$\displaystyle \int_0^4 \int_{\sqrt{4x-x^{2}}}^{2\...
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How to find out the order of integration of a combination of iid random series?
$\varepsilon_{1j}, \varepsilon_{2j}, \varepsilon_{3j} ∼ iid(0; 1)$ and they are three independent processes.
$ y_{t} =x\sum_{i=1}^{t}\sum_{j=1}^{i} \varepsilon_{1j} + y\sum_{i=1}^{t} \varepsilon_{2i}...
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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 ...
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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 ...
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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 $\sin x \sin y=\sin z$, ...
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Triple Integrals - When does order of integration not matter?
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)=...
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