Questions tagged [multiple-integral]
For questions regarding computation and results related to integrals in at least 2 variables.
1,521
questions
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Define interval $M$ for inequality $(x^2+y^2)^2\leq2y^3$ and calculate integral $\iint_M1dxdy$
I first thought I could use polar substitution where $\phi$ would be $0\leq\phi\leq\pi$, but I couldn't figure out the bounds for the radius.
I think it would be better to just define it normally ...
1
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0
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34
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changing order of integration, double integral
Determine $$I:=\int_{[0,\infty)}\int_{[y,\infty)}\sin\left(\frac{\pi y}{2x}\right)\frac{e^{-x}}{x}\,d\lambda(x)\,d\lambda(y).$$
I tried as follows: I want to calculate $$\lim\limits_{n\to\infty}\int_{[...
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1
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26
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wedge volume problem
Find the volume of the wedge cut from the first octant by the cylinder
$z = 12 - 3y^2$ and the plane $x+y=2$.
What I did- sketched parabola and repeated in all of x axis, drew the plane and found the ...
4
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65
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Why is $\int_{0}^{2\pi} \int_0^{2\pi} \frac{\ln(21-4(\cos x+\cos y+\cos(x+y)))}{2\ln(9/2)}\frac{dx}{2\pi} \frac{dy}{2\pi}$ almost $1$?
Consider the function
$$ f(x,y) = \frac{\ln(21-4(\cos(x)+\cos(y)+\cos(x+y)))}{2\ln(9/2)} $$
Its average value is awfully close to unity:
$$ \int_{0}^{2\pi} \int_0^{2\pi} f(x,y) \frac{\mathrm dx}{2\pi} ...
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0
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23
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Using multivector residue theorem to evaluate multiple integrals
I recently started learning about Geometric Algebra and Geometric Calculus. Since the residue theorem can be generalized for multivector functions I wondered if one could use it to evaluate certain ...
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2
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52
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Find volume of solid bounded by given surfaces. $z=a+x,z=-a-x,x^2+y^2=a^2$
Find volume of solid bounded by given surfaces.
$$z=a+x, \qquad z=-a-x, \qquad x^2+y^2=a^2$$
This is the solid. We can find volume of solid that has positive $z$ value and multiply by $2$. And for ...
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18
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Solution to multidimensional gaussian integral with power factor
I would like to figure out for the derivation for a tensor ABCD optical law was derived for a flattened Gaussian beam in the following reference:
https://www.sciencedirect.com/science/article/pii/...
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16
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Triple integral set up using cylindrical coordinates
Set up an integral in cylindrical coordinates to evaluate $\iiint_{E} x y d V$ where $E$ is the region enclosed by the cone $z=2-\sqrt{x^{2}+y^{2}}$, the cylinder $x^{2}+y^{2}=1$, and the $x y$ plane.
...
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42
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What does $d^n\textbf{x}$ mean in this context?
I found the following on Wikipedia.
Integration over more general domains is possible. The integral of a function $f$, with respect to volume, over an $n$-dimensional region $D$ of $\mathbb{R}^{n}$ ...
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30
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Does multiple integral have the second definition?
I have noticed that there are two definitions in Riemann integral(not multiple).
Definition 1.
For all $\epsilon > 0$, there exists $\delta > 0$ such that for any tagged partition $x_0$, ..., $...
2
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1
answer
181
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What is the average distance between two points on a unit square without using a pdf?
I was trying to solve this question through a different method and I am not getting the right answer.
So my approach was first to figure out what would be the average distance between two points on a ...
1
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0
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32
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double integral showing equal to 1
The question asked is as follows:
Given that the nonnegative function $g(z)$ has the property that $\int_{-\infty}^\infty g(z)dz=1$ show that $f(x,y)=\frac{g(\sqrt{x^2+y^2})}{\pi\sqrt{x^2+y^2}}$ for ...
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How to find bounds for double integration over a region?
I don't quite understand how to do double integration for the joint probability density function through looking at a graph. Which comes from this question:
Let $X$ and $Y$ have the joint pdf $f_{X,Y} ...
2
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1
answer
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Evaluation of double integral $I=\int_{-1}^{1}\int_{0}^{x^2} \sqrt{x^2-y}\,dy\,dx$
I have the following double integral before me:
$$I=\int_{-1}^{1}\int_{0}^{x^2} \sqrt{x^2-y}\,dy\,dx$$
I got the answer of this integral as $0$ working in the following manner:
$$I=\int_{-1}^{1} \...
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41
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Show that $\int_0^1 dx\int_0^1 f dy=1$, but $\int_0^1 dy\int_0^1 f dx$ does not exists.
If $A=\left\{0\leq x\leq 1;0\leq y\leq 1\right\}$ and $f:A\to \mathbb{R}$ is defined by
$$f(x,y)= \begin{cases} 1 & \text{if } x\in \mathbb{Q} \\ 3y^2 & \text{if } x \in\mathbb{Q^c}\end{cases}
...
2
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0
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Show example that if $\phi$ is not injective then the change of variables might yield different result
I need to find non-injective parameterization to some domain such that :
$$\int_D f(x,y) dxdy \ne \int_E (f\circ \phi)(u,v)J(u,v) dudv$$
The example I found turns out to be wrong. (The integral is the ...
2
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2
answers
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Double integral and change of variable [closed]
$$
\iint_D \left(x^2-y^2\right)\ dxdy
$$
over $D$ which is bounded by region enclosed by the four curves $y = x, y = x + 1, xy = 1$ and $xy = 2$ in the first quadrant.
What will be a suitable change ...
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1
answer
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Evaluating the polar coordinate integral $\int e^{k\mu^Tx}dx$ when $c(k)\int e^{k\mu^Tx}dx = 1$ and $x \in \mathbb{S}^{p-1}$
Let $x \in \mathbb{S}^{n}$ be a point on the unit $n$-sphere with coordinates $\begin{cases} x_1 &= \cos(\theta_1).\\
x_2 &= \sin(\theta_1)\cos(\theta_2).\\
x_3 &= \sin(\...
0
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1
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65
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Calculate $\int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{|x-y|}} \mathrm{d}x\mathrm{d}y$
I am asked to evaluate de following integral:
$$\int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{|x-y|}} \mathrm{d}x\mathrm{d}y$$
My attempt.
Note that
$$
\frac{x-y}{|x-y|} = 1, \text{ if } x-y > 0
...
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0
answers
19
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definite integral with inner integral depending on outer integral variable and Gauss quadrature
I'm trying to figure out how to solve this definite integral:
$\int _0^z\left(x(l)+ \int_0^l\left(w(r)dr\right)\right)dl$
As you can see the $z$ is the range of integration (constant) and the inner ...
0
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1
answer
27
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Volume between cone and sphere with triple integral
I am trying to find the volume of $E=\{(x,y,z)\in\mathbb{R^3}:\sqrt{\frac{x^2+y^2}{3}}\leq z\leq \sqrt{4-x^2-y^2}\}$ and I would be grateful if someone could check out my work. Comments are welcome. ...
0
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15
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Applied mathematics and Physics applications of improper multiple integrals
I'm interested in multiple improper integrals, and I discovered new theorems. Can I know some applications especially in physics or in applied mathematics, where we can use multiple integrals....
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1
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61
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Triple integral in Cartesian Coordinates
Let $R$ be the region in the first octant bounded by a surface $F(x,y,z) = 0$ and the
coordinate planes. The
projection of $R$
on the $xy$-plane is bounded by the coordinates axes and the curve $y = ...
2
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3
answers
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$\iint_D(x^2-y^2)dxdy$ with D enclosed by $y=\frac2x$, $y=\frac4x$, $y=x$, $y=x-3$?
I have been presented with the following problem:
Calculate the double integral $$\iint_D(x^2-y^2)dxdy$$ where D is the area enclosed by the curves $y=\frac2x$, $y=\frac4x$, $y=x$, and $y=x-3$.
Here's ...
0
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0
answers
63
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What's the derivative of the following integral?
So given a fixed $r \in \mathbb{R}$, I have the following function:
$$F(\theta) = \frac{1}{r}\int\limits_0^{r}f(a+ p \cos θ, b +p\sin θ)\ \mathrm dp$$
So $F$ is basically calculating the average value ...
0
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1
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44
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Improper integral of $\frac{1}{|x|^p+|y|^q}$
Question
Show that
$$
\int_{D}\frac{1}{|x|^p+|y|^q},
$$
where $D=\{(x,y)\in\mathbb{R}^2\ |\ 0<|x|+|y|\leq1\}$, exists when $\tfrac{1}{p}+\tfrac{1}{q}<1$.
Attempt
So far, I've been trying to ...
2
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1
answer
78
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Trouble with a triple integral on a region bounded by a sphere and two planes
I would like to compute the integral $\int_A zdzdydx,$ where $A$ is the region bounded by the sphere $x^2+y^2+z^2=R^2,$ plane $\frac{x}a+\frac{y}b=1$ and coordinate planes (which doesn't contain the ...
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1
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How to calculate the integral of $\vec{a}\cdot\nabla(\nabla\cdot\vec{a})$?
I want to calculate the following integral:
$$S = \int\vec{a}\cdot\nabla(\nabla\cdot\vec{a})\,\text{d}V$$
I tried to calculate the integral of the $i$-th term, i. e., $\int a_i\partial_i(\nabla\cdot\...
10
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2
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393
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Entropy of fair but correlated coin flips
Consider the joint distribution, $p(\xi_1,...\xi_N)$, with components defined as $\xi_i=\mathrm{sign}(x_i)$, with $(x_1,...,x_N)\sim\mathcal{N}(0,\Sigma)$ with
$
\Sigma_{ij}=\delta_{ij}+(1-\delta_{ij})...
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Integrating on a subspace of a scalar field = integrating on the orthogonal subspace of the Fourier transform
Suppose you have some function $f: \Bbb R^n \to \Bbb R$. You also have an injective linear transformation $A$ mapping from $\Bbb R^m \to \Bbb R^n$, with $m < n$, so that the image of $A$ is some $m$...
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How How can I solve this integral? [duplicate]
I try with spherical coordinates, some tip?
$\int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1} \frac{x^2}{x^2+y^2+z^2} dx dy dz $
0
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0
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Evaluate $\iiint \frac{1}{s \left(- x + y\right) \left(- b z + x y\right) \left(x \left(r - z\right) - y\right)}\, dx\, dy\, dz$
Consider the following integral where $s,r,b \in \mathbb{R}_{>0}$ are constants and $x,y,z\in \mathbb{R}$ are independent variables
$$I = \iiint \frac{1}{s \left(- x + y\right) \left(- b z + x y\...
2
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2
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Double integral of $1/(x^2+y^2)$ restricted to $x^2+y^2\leq2$ and $x\leq1$
Find $$\iint_D \frac{1}{(x^2+y^2)^2}dA$$ where $$D = \left\{ (x,y): x^2 + y^2 \leq 2 \right\} \cap \left\{ (x,y): x \geq 1 \right\}$$
Because of the prevalence of $x^2+y^2$ terms here, I figured we ...
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1
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36
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(Multiple Integral) The mass of a bullet shape, given its density and equation for the curved top
I'm trying to calculate the mass of a bullet-shaped object; a cylinder and a curved top, where only the density and equation for the round top is known.
bullet-shaped object is formed from a cylinder ...
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0
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16
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Boundary for $\phi$ in Double Integrals using Polar coordinates
Suppose we have polar coordinates $(r,\phi)$, where $r≥0$ and $0≤\phi<2\pi$.
I have a question about the upper boundary for $\phi$ when evaluating some double integrals:
Why do we write $0≤\phi≤2\...
2
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1
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Range of $\phi, \theta$ in $\int_0^{\pi/4} \int_0^{\pi/2} \int_0^{2\sin\phi \sin\theta} \rho^3\sin\phi \sin\theta d\rho d\theta d\phi$
The question: A solid bounded by the (y,z)-plane, the (x,y)-plane, the cone $x^2 + y^2 = z^2$, and the surface $x^2 + y^2 + z^2 - 2y = 0$. Suppose a density of a chunk of metal of the shape of this ...
3
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2
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Rewrite the integral in the order dy dz dx
Rewrite the integral $$\int_0^3 \int_0^{9-y^2} \int_\frac{y}3^1 f(x,y,z) dx dz dy $$ as an interated integral in the order dy dz dx.
I have trouble visualizing if my answer is the correct iterated ...
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1
answer
35
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Proofing the identity of double integrals over symmetric function
In one of my physics classes we had given a double integral over a function $f$ in two variables:
$$G=\int_{x\in D}\int_{y\in D}f(x,y)\ \mathrm{d}y\ \mathrm{d}x$$
The function $f$ is symmetric in its ...
0
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0
answers
40
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Volume of a piece of a sphere
Given the sphere of radius two centered at the origin, you can slice a piece off by imposing the conditions that $z\ge 0$ and $y\ge 1$. I've been instructed to find the volume of this piece using ...
1
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1
answer
79
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Integral on a rotational solid
Compite the integral $$\int_V\frac{4-z^2}{(x^2+y^2)^3}\mathrm dx\mathrm dy\mathrm dz,$$ where $V$ is the solid enclosed by the paraboloids $x^2+y^2=z,x^2+y^2=2z$ and cones $x^2+y^2=(z-2)^2,x^2+y^2=4(z-...
1
vote
1
answer
91
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Convolution with continous 2 random variables. The integration formula seems quite strange for me.
The main theme of this post is painted with red below. Please scroll down a bit to see it.
$$X,Y:=\text{2 independent continuous random variables}\tag{1}$$
$$\text{These random variables follow the ...
0
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1
answer
35
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A Surface integral over the intersection of a plane and a sphere
I've been banging my head against this thing for the last 4 hours to no avail
Evaluate the integral $ \iint_S curl \vec{F}\cdot d\vec{S}$ where $S$ is the portion of the surface of the sphere defined ...
2
votes
0
answers
36
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Express symmetric double integral in terms of single integral
Is it possible to express the following double integral using only a single integral?
$$
\int_0^\infty \int_0^\infty \frac{f(x)f(y)}{ia-x-y}dxdy,
$$
where $a$ is real.
Are there any general theorems ...
2
votes
3
answers
156
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Finding the volume with triple integrals
I want to find the volume of a function described by:
$$ G= \{(x,y,z)|\sqrt{x^2+y^2} \le z \le 1, (x-1)^2+y^2 \le 1\}$$
This question can be best solved in cylindrical coordinates. So if I follow that ...
1
vote
2
answers
48
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Is double integration an easier way to find volume of rotation?
AP Calculus BC student here,
One of the most hated topics from Calculus 1 & 2 is often the disk method, washer method, and the shell method.
Disk Method = $\pi \int [f(x)^2]dx$ (rotate x-axis)
...
1
vote
1
answer
69
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Transformation of limits after variable substitution in a double integral
Let $$f(x, y)=\left\{\begin{array}{ll}\frac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0)\end{array}\right.$$Show ...
1
vote
2
answers
41
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How can solve this surface integral without stokes theorem?
A 3D field vector field $F$ for a dynamic system is given as
$$\vec F(x,y,z)=x\hat i+y\hat j+z^4\hat k$$
Find $$\iint \vec F\cdot d\vec S$$
where $S$ is the part of the cone $z=\sqrt{x^2+y^2}$ below ...
-1
votes
2
answers
52
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triple integral on cone
Hello everyone I have to calculate
$\int\int\int (x^2+y^2+z^2)^\alpha dxdydz$ on the cone $z=\sqrt{x^2 + y^2}$ which has a height of 1 and base circumference $x^2+y^2=1$.
$\alpha >0$.
I considered ...
2
votes
3
answers
57
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Limits of $u$ and $v$
I am trying to evaluate a double integral $$I=\int_{0}^2\int_{0}^{2-x}(x+y)^2e^{\frac{2y}{x+y}}dydx$$
I used the transformation $$x+y=v, y=uv$$
That is $$x=v(1-u), y=uv$$
We get the Jacobian as:
$$J=\...
0
votes
1
answer
70
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Double integral $ \int_{0}^{\alpha}dx\int_{0}^{\alpha}dy\frac{\ln|x-y|}{\sqrt{xy}}$ [closed]
How do I show the following:
$$
\int_{0}^{\alpha}dx\int_{0}^{\alpha}dy\frac{\ln|x-y|}{\sqrt{xy}}=4\alpha(\ln(\alpha)+2\ln(2)-3)
$$
where $\alpha>0$. The integral has arisen as I've been studying ...