Questions tagged [iterated-integrals]
This tag is for questions relating to iterated integrals. In calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example, $~f(x,y)~$ or $~f(x,y,z)~$) in a way that each of the integrals considers some of the variables as given constants.
127
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Does Fubini's theorem apply on this infinite region?
I came across the following example for a triple integral:
Find the volume of the region bounded by hyperbolic cylinders:
$$ xy = 1 \quad , \quad xy = 9$$
$$ xz = 4 \quad , \quad xz = 36 $$
$$ yz = 25 ...
- 43
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1
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Can I change the order of integration when the upper limits are infinite?
I'm trying to solve this :
$$\frac{d}{dy}\int_{y}^{\infty} \int_{f(y)}^{\infty} (g(y)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$$
In this case, can I change the order of Integration ? I will get this :
$$\frac{d}{...
- 25
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0
answers
20
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Two double integrals (in terms of polar) producing the same result?
I was working on a problem where I had to convert the region into polar equations in order to find the double integral. The region was
$$r = 2\cos(\theta)$$ and after solving the problem I realized ...
- 203
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votes
2
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Solve $\int_{0}^{1}\int_{0}^{1-y} xy\sqrt{(1-x-y)}dxdy$ [closed]
Calculate the double integral
$$∫∫xy\sqrt{1-x-y}\,dxdy$$
where the domain is $D=\{(x,y):x≥0,y≥0,x+y≤1\}$
I think the range is $0≤x≤1$
and $0≤y≤1−x$
. Is it correct? I understands that this problem ...
1
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2
answers
70
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Stokes' theorem for the vector field $\vec{F}(x,y,z)=(y,-x,xz)$ on the surface $z=9-x^2-y^2$
Verify Stokes' theorem for the vector field $\vec{F}(x,y,z)=(y,-x,xz)$ on the surface $z=9-x^2-y^2$ with $z\geq 0$.
So I've tried finding the parametrisation which is $$\vec{r}(\theta,r)=(r\cos\theta,...
- 111
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47
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How to denote an iterated integral with respect to a set of variables?
Let $x := (x_1, \dots, x_n) \in \mathbb{R}^n$ be a vector of states with $n \in \mathbb{N}$ dimensions in the space of real numbers and let $f, g : \mathbb{R}^n \to [0, 1]$ be some differentiable ...
- 119
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Find the area of the plane region defined by $x^{2/3}+y^{2/3}≤ 1$ , $x≥0$ , $y≥0$ using a change of variables
Find the area of the plane region defined by $x^{2/3}+y^{2/3}\leq 1$ , $x\geq 0$ , $y\geq 0$ using the change of variables $x=r\cos^3 \theta$ , $y=r\sin^3 \theta$
The teacher used polar coordinates ...
- 348
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1
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Double integral over a half-plane
Let $$f(x,y)=\frac{1}{N_0\pi}\exp(-\frac{(x-a)^2+(y-\frac{b}{2})^2}{N_0})$$ I want to compute $$I=\int_{D_2}f(x,y)d\vec{x}$$ where $D_2$ is the half-plane above the line $2x-y=\frac{a}{4}$: $$D_2=\{(x,...
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Convert the iterated integral $ \int_{0}^{6}\int_{0}^{\sqrt{36-x^2}}e^{-x^2-y^2}dydx $ into polar integral
$$
\int_{0}^{6}\int_{0}^{\sqrt{36-x^2}}e^{-x^2-y^2}dydx
$$
I tried converting it into polar coordinates using the following equations
$y = \sqrt{36-x^2} \implies y^2 + x^2 = 36 \implies r = 6$
$0=6\...
0
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1
answer
20
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Help with formula on linear nested conditioned expectations
I am working on a nested integration problem and want to develop an efficient estimator for said problem.
The problem has the form:
$\mathbb{E}_x\left[F(x,\mathbb{E}_y\left[G(y, x))\right]\right] $ ,
...
- 3
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39
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If two variable function on a rectangle is continuous, should the iterated integrals must exist?
Let $f(x,y)$ be a two variable function on a rectangle $A$.
There are cases when iterated integral doesn't exist but $f$ is integrable on $A$.
But are there cases when iterated integral doesn't exist ...
- 536
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1
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155
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Please clear my mathematical physics confusion regarding integration.
EDIT: If multiple Riemann integral is not ok, then please consider Lebesgue integral.
Consider Cartesian coordinate system:
Let there be a cubic charge $V'$ of side $a$ units with uniform charge ...
- 536
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1
answer
47
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Is the tower property true conditional on vector?
Let $X|\theta \sim F$ with $\theta$ a random vector and $X$ a random variable. Is it true that
$$
E[X] = E[E[X|\theta]]
$$
even though $\theta$ is a vector? If so, can one say that if $X|\mu,\sigma \...
- 355
1
vote
3
answers
219
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Iterated conditional probability notation
I'm currently self-studying Andrew Gelman's book "Bayesian Data Analysis" third edition. At the page 41, they write:
$E(\tilde{y}|y)=E(E(\tilde{y}|\theta,y)|y)$
I am ok with multiple ...
- 353
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0
answers
40
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Confusion over changing the bounds of an iterated integral
I have the following:
The sum of the iterated integrals $$ \int_{-2}^{-1}\int_{-\sqrt{y+2}}^{\sqrt{y+2}} f(x,y)\, dx\, dy + \int_{-1}^{2}\int_{y}^{\sqrt{y+2}} f(x,y)\, dx\, dy$$ is equal to:
a) $\...
- 2,677
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votes
2
answers
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Volumen of the intersection of a cone and a tangent cone.
the professor of our integration class gave us the following exercise as an assignment:
Determine the measure in $\mathbb{R}^3$ of $A=\{(x,y,z) \in \mathbb{R}^3 ,x^2+y^2+z^2-2x+2z\leq0\leq x^2-y^2-z^2 ...
- 573
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0
answers
64
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A polynomial sequence by iterated integration
I have been inspired by this question to think about the polynomial sequence $a : \mathbb{N} \rightarrow \mathbb{Q}[x]$ defined by $$a_n(x) = \int_0^{1{-}x}dy\,a_{n{-}1}(y),\qquad a_0 = 1.$$ The first ...
- 1,286
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area of one rose petal using iterated integral but the outer integral is of radius
I have a Rose leaf, described by the equation r=a*sin(3θ), from θ=0 to $\frac{\pi}{6}$.
I need to make an iterated integral of this area.
It's easy when the outer ...
- 11
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0
answers
20
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Using iterated integrals to get the area with strict conditions
We were tasked to use iterated integrals over this area shaded in yellow here:
I'm confused as to how to approach this as I've only seen ones that encompass the whole space inside the closed shape ...
- 1
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1
answer
47
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How to calculate this iterated integral:$\int_0^1dy\int_0^y dx\int_0^x \frac{e^z}{1-z}dz$?
Is there any trivial way to calculate this iterated integral,$\int_0^1dy\int_0^ydx\int_0^x\frac{e^z}{1-z}dz$? I've already solved this by taking series expansion of $e^x$ into the integral,but I ...
- 63
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vote
0
answers
42
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Writing the iterated expectation with a single integral
I would like to write the expected value of $c(x)$ where $x$ is sampled from a distribution $\gamma(x|m)$ and $m$ is sampled from another distribution $\omega(m)$. Here, for any fixed $m$, the ...
0
votes
1
answer
42
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How we solve iterated integrals such as this one?
Here is the problem
$(\textbf{20}\text{ points})$ Calculate the following iterated
intergals:
$$\text{a. } \int_0^2\mathrm dx
\int_{-1}^1\big(3x^2-(x+y)e^y+xy^3\big)\ \mathrm dy,$$
I can do the ...
- 1
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3
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$\int_a^x \int_a^t f(t_1) dt_1 dt = \int_a^x (x-t) f(t) dt$?
A quick question on confusion on some iterated integral in the text I am studying :
Defined in Special Functions by Roy :
How $\int_a^x \int_a^t f(t_1) dt_1 dt = \int_a^x (x-t) f(t) dt$ is true? ...
user200918
7
votes
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answer
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Ramanujan: If $\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx$, then $\frac\pi2\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty\psi(q,x)\psi(p,nx)dx$.
Apparently, the following appeared in Ramanujan's first letter to GH Hardy.
If $$\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx,$$
then $$\frac{\pi}{2}\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty \psi(q,x)\psi(...
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Can we write the right hand side as a one integral formula? (A lecture on multivariable mathematics by Theodore Shifrin on YouTube)
I am watching a lecture on multivariable mathematics by Theodore Shifrin on YouTube.
I want to know the answer to this problem.
My answer is the following. My answer uses two integral formulae.
$$\...
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Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates.
Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates.
I could say that: $0\leq 1, 0\leq x\leq z, 0\leq y \leq \...
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Calcuate the integral $\int_1^{-1}dx\int_x^{2x}e^{x+y}dy$
Calcuate the integral $\int_1^{-1}dx\int_x^{2x}e^{x+y}dy$
Here is my working out:
Split the integral into their respective derivate parts: -
$$\int_1^{-1}e^xdx\int_x^{2x}e^ydy$$
The RHS is given as :
$...
- 361
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0
answers
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Iterated integrals, range is given by inequalities
This problem is from mathematical statistics 8ed by hogg
Let $C = \mathbb R^n$ For $A$ in $C$ define the set function
$$ Q(A) = \idotsint\limits_A dx_1 dx_2 \cdots dx_n$$
If $B = \{(x_1, x_2, \dots, ...
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2
answers
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Changing integration order in an iterated double integral
I am trying to solve the integral below (equation 1) by changing the integration order from dydx to dxdy:
$$ I = \int_{\frac{2}{\sqrt{5}}}^1\int_{\sqrt{1-x^2}}^{\frac{x}{2}}f(x,y) \mathrm dy \mathrm ...
- 41
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2
answers
148
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Iterative integration with indicator function
I have an exercice in which the result
$$
\int_{[0,1]^n}\mathbf{1}_{\{0<x_n<\ldots<x_2<x_1<1\}}\prod_{k=1}^{n}\mathrm{d}x_k=\frac{1}{n!}
$$
is given in a development like it's obvious, ...
- 175
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3
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Given a general region, find the double integral bounded between $y = x$ and $y=3x-x^2 $
$$ J = \iint_R (x^2-xy)\,dx \,dy, $$
Suppose region R is bounded between $y = x$ and $y=3x-x^2 $
My attempt using vertical integration:
$$ \int^{x=2}_{x=0} \int^{y=3x-x^2}_{y=x} \left({x^2-xy}\right)...
- 418
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1
answer
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How to construct double integral to find the area, given $\mathsf{D}= { \left( x,y \right): y \geq x^2-1 , y \leq 2x+2 , y \leq 3}$
I don't understand this topic honestly.
I thought there is supposed to be two types of ranges for x and y.
$$\therefore \space \mathsf{for} \space \space x^2-1 \leq 3 \space ; \space x \in [-2,2] \...
- 33
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votes
1
answer
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Infinite sum of iterated integrals of matrix products
Edit: Discussion moved to Mathoverflow at https://mathoverflow.net/questions/395085/infinite-sum-of-iterated-integrals-of-matrix-products
The problem:
Let
$$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \...
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2
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Integral of $\int ^{5}_{1}\int ^{\sqrt{x-1}}_{y=0}{ye^{(x-1)^2}}dydx$
$\int ^{5}_{1}\int ^{\sqrt{x-1}}_{y=0}{ye^{(x-1)^2}}dydx$
First, I changed the integration from $x$ to $y$.
However, I get $\int ^{5}_{1}\dfrac{\left( x-1\right) }{2}e^{\left( x-1\right) ^{2}}dx$ and ...
user937947
1
vote
2
answers
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Reverse the order of integration of $ \int_{-1}^1 \int_{x^2}^1 \sqrt{y}\ dydx$.
I am computing the following integral $$ \int_{-1}^1 \int_{x^2}^1 \sqrt{y}\ dydx$$ and I want to verify the exchanging of order of integration. The region is described as $-1\leq x \leq 1, x^2\leq y \...
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0
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A question about iterated integral
In a circumstance I had to show that
$$
\int_0^t \int_0^x \int_0^y f(z)\,dz\,dy\,dx=\frac{1}{2}\int_0^t f(z)(t-z)^2 \,dz
$$
, where $t$ is a constant.
I succeeded in doing so by integration by parts. ...
- 411
1
vote
1
answer
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$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$
I would like to solve the given integral:
$$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$$
The integral is doable just using the regular iterated integral with $x$ and $y$. We integrate with respect ...
- 2,012
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votes
0
answers
37
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Greens Theorem & does the perimeter of a double integral over a region converge to the perimeter of the original region?
When I've seen double integrals presented, usually its visualized as adding a bunch of small rectangular dA elements along the region. It feels pretty reasonable that this converges to the area.
What ...
- 1
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votes
1
answer
36
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Questions using Iterated Integrals.
so Ive just been introduced to the idea of iterated integrals and im finding it hard to work out how to complete questions on this subject and was wondering if anyone could help.
So if I have an ...
- 353
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1
answer
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how to prove that two iterated integral are different
We have the following function: $$f(x,y)=
\begin{cases}
1, & 0 \leq x-y \leq 1\\
-1, & 0 \leq y-x \leq 1 \\
0, & \text{other cases}
\end{cases} $$
and I need to prove that $...
- 159
1
vote
1
answer
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How do I get the normal domain from an inequality condition (application of Fubini's Theorem)?
I want to compute the volume of the 3-dimensional unit ball.
$A := \{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \leq 1$
From this I want to get the normal domain as:
$A = \{(x,y,z): x \in [-1,1], y \in ...
- 139
1
vote
1
answer
235
views
Find the surface area of the sphere inside the cylinder
The given equations are of a sphere and cylinder respectively
$$x^2+y^2+z^2=400$$
$$x^2+y^2=256$$
Solving the sphere equation for $z$ yields
$$z=\sqrt{400-x^2-y^2}$$
Now to find $ds$ we take the ...
- 1,073
2
votes
2
answers
367
views
Problem with Infinite number of iterated integrals
I have been working on a problem from a past high school Math Competition and I have been stumped. I am asked to compute
$$\lim_{n\rightarrow\infty} \int_{0}^{1}\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}\...
- 57
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vote
0
answers
202
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Converting an iterated integral into a **sum** of iterated integrals
Question: Suppose $f$ is integrable. Express $\int^1_0 \int^{2y}_y \int^{x+y}_0 f(x,y,z)\ dz\ dx\ dy$ as a sum of iterated integrals in the order $\ dx \ dy \ dz$.
My Attempt: Since $z=x+y$, $z_{max}=...
- 383
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votes
1
answer
39
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continuity of a function and iterated integrals
Let $f:[0,1]\times [0,1] \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ defined as :
\begin{equation}
\label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande}
f(x,y) = \left\{
\...
0
votes
1
answer
44
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I have a question on Lebesgue Iterated Integrals
Let $(X, S, \mu)$ be a finite measure space. Suppose $g: X \rightarrow \mathbb{R}$ is integrable and that $g(x) \geq 0 \ \forall x \in X$. Let $A = \{(x,y) \in X \times [0,\infty) : 0 \leq y < g(x)...
- 103
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votes
1
answer
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how to use iterated integral to find area of a region? [closed]
Use iterated integral to find area of a region bounded by $y=\sqrt{x}, y=3, x=0$, and $x=4$. sketch the region.
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votes
3
answers
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Why are these two inequalities not the same even though they use the same equation?
I just don't get it, like at all.
$U_{n}$ is an iteration defined on $\mathbb{N}$, BTW.
The question was:
$$\begin{align}
U_{n+1} &=
\frac{8U_n - 8}{U_{n} + 2} = 8 - \frac{24}{U_{n} + 2}\\
...
- 423
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votes
0
answers
39
views
Am I putting the right limits of the integral?
Let:
\begin{align}
r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\
s&=a\\
t&=p\\
f(r) &= \text{continuous function of } r\\
g(s) &= \text{continuous function of } s\\
\end{align}
...
- 1,111
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votes
1
answer
52
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Proving the non-integrability of the following function on $(0,1)\times(0,1)$
So, basically, I've been trying for a while to prove that the function $f(x,y)=\frac{x-y}{(x+y)^3}$ is not integrable in $(0,1)\times(0,1)$. That is, I want to prove that the integral of it's absolute ...
user246480