Questions tagged [multiple-integral]

For questions regarding computation and results related to integrals in at least 2 variables.

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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 ...
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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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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 ...
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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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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 answers
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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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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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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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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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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 $x_0$, ..., $...
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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 ...
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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} ...
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2 votes
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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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} ...
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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 ...
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2 votes
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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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(\...
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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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  • 139
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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 ...
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  • 117
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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. ...
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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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61 views

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 $y = ...
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2 votes
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$\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 ...
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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 ...
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1 answer
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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 ...
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2 votes
1 answer
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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 answer
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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\...
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10 votes
2 answers
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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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1 vote
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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 $
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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\...
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2 votes
2 answers
94 views

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 answer
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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 answers
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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\...
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2 votes
1 answer
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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 ...
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3 votes
2 answers
101 views

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
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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 ...
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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 ...
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1 vote
1 answer
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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-...
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1 vote
1 answer
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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 ...
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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 ...
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2 votes
0 answers
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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 ...
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  • 430
2 votes
3 answers
156 views

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 ...
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1 vote
2 answers
48 views

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) ...
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1 vote
1 answer
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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 ...
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1 vote
2 answers
41 views

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 ...
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-1 votes
2 answers
52 views

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 ...
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2 votes
3 answers
57 views

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=\...
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1 answer
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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 ...
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