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Questions tagged [multiple-integral]

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

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Question about Lemma 19.1 in Munkres' Analysis on Manifolds

In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
studyhard's user avatar
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Stokes’ theorem question for cylinder-like surface with two disjoint boundaries

Q: Consider $S=\{(x,y,z)\mid x^2+y^2+z^2=25,\; 0≤z≤4\}$ and the vector field $\vec F(x,y,z)=(y,-z,x)$. Use Stoke’s theorem to calculate $\iint_S(\nabla \times \vec F )\cdot \vec n \; dS$ $\vec n$ is ...
Jason Xu's user avatar
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-4 votes
2 answers
151 views

Theorem 16.5, Munkres' Analysis on Manifolds [closed]

In Munkres' Analysis on Manifolds, page 142 Theorem 16.5 it states: $$\int_{D}f\leq\int_{A}f$$ at the end of that page. Here, $D=S_{1}\cup\cdots\cup S_{N}$ is compact since $S_{i}=Support\phi_{i}$ and ...
studyhard's user avatar
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Parametric Curve

Evaluate the following double integral: $I=\iint_{D}^{}y dydx$ with the region $D$ defined by $\left\{\begin{matrix} x=R(t-sint) & & \\ y=R(1-cost), y=0 & & \\ 0\leq t\leq 2\pi&...
SpyroK's user avatar
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Interchanging differentiation and double integral

Consider the double integral $$I(a)=\int_0^1\int_0^1 f(x,y,a)dxdy$$ What are the conditions needed so that we can differentiate under the integral sign of this double integral? That is when do we have ...
Max's user avatar
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2 votes
2 answers
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Why this simple volume problem in Multivariate Calculus seems to have an anomaly?

Find the volume of the solid inside the cylinder $x^2+y^2-2ay = 0$ and between the plane $z = 0$ and the cone $x^2+y^2 = z^2$. I tried solving this problem as follows: Equation of the cylinder $x^2+(y-...
Thomas Finley's user avatar
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A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.

Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$ Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
Thomas Finley's user avatar
1 vote
0 answers
16 views

Computing the multiple-integral of a $n$ variable symmetric function

I'm interested in evaluating the following integral analytically, numerically, or using a combination of both. Let $$ I_n(\rho) = 2^n\int_{\mathbb{R}_+^n} \sqrt{1+ \rho^2 \left(\prod_{k=1}^n x_k^{-2}\...
P.S. Dester's user avatar
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere

This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$. Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
Hussain-Alqatari's user avatar
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Multiple integral with constraint

Let $\mathbf{x}\in\mathbb{R}^n$ and consider the multidimensional integral $$ I=\int \mathrm{d}^n\mathbf{x}\,\Phi[\phi(\mathbf{x})] $$ where the dependence of the integrand on $\mathbf{x}$ enters only ...
CW279's user avatar
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$

I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
A Math Wonderer's user avatar
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How has this integral been written as a line integral?

I am right now self-learning Green's functions for partial differential equations and I am stuck at the very last step of deriving the adjoint operator. To begin, the PDE of interest is $\textbf L u = ...
ishan_ae's user avatar
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Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$

I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$ Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
Max's user avatar
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Probability densities with conditions - how to find the distribution function

I have two probability density functions where i need to find the distribution function. The first function is $$f(x,y)= \begin{cases} \frac{x}{y} & \text{for $0\leq x\leq y\leq c$}\\ 0&\text{...
mscr's user avatar
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Triple Integral - Use symmetry for center of mass question?

I am unsure when to use symmetry with triple integrals. Can I use symmetry for this centre of mass question? $E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; ...
user41592's user avatar
  • 143
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1 answer
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Triple integral (mass) - setting up region between planes and parabolic cylinder

I am trying to set up the following triple integral using the xy plane. $E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; \quad \rho(x, y, z)=8$. I set up ...
user41592's user avatar
  • 143
2 votes
1 answer
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Evaluation of the given line integral

Question: Evaluate $\int_{C}$B.dr along the curve $x^{2}$+$y^{2}$=1,$z$= 1 in the positive direction from (0,1,2) to (1,0,2);given B= (xz²+y)i+(z-y)j+(xy-z)k The question itself is easy,but I don't ...
The Sapient's user avatar
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Indexes of nonhyperbolic equilibrium points in planar vector fields

There is a well-known theorem in dynamical systems stating that if $\gamma$ is a "sufficiently nice" closed curve (continuous, piecewise smooth, nonconstant function from $[a,b]$, say $[0,1]$...
Boris Dimitrov's user avatar
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1 answer
37 views

Changing the integration limits of a triple integral

I have a triple integral of the form $$ \int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_1} dt_3\ f(t_1,t_2,t_3) $$ and I want to transform it to the form $$ \int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3\...
SrJaimito's user avatar
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1 vote
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the role of supp(f) in the change of variables formula

I learned that the change of variables formula,but I can't understand why we require that supp(f) is contained in φ(U).What will happen if I remove this condition? Theorem — Let U be an open set in ...
Sam's user avatar
  • 89
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1 answer
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Identify the special function with this sum and integral form

There is a multivariate generalization of the Bessel function that has both a sum and integral form. Both are functions of a vector $\mathbf{0}\leq\mathbf{x}\in\mathbb{R}^n$, with parameters defined ...
Victor V Albert's user avatar
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A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists

A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists In Munkres' Analysis on Manifolds, page 94 Theorem 11.2 it states: Divide ...
studyhard's user avatar
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Null value for a double integral

This question is correlated with this: Clarifications on the solution of a double integral: $\iint_X\frac{x^2y}{x^2+y^2}dxdy$. If I consider $$ \int_1^{2\cos\vartheta}r^2dr\underbrace{\left(\int_{-\...
Sebastiano's user avatar
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0 answers
21 views

Integration Problem with Proof that 4 Random Variables are Independent Given Their Joint pdf

I have a problem where I need to show that 4 random variables are independent given their joint pdf. I just want to make sure I have the right approach: $W, X, Y \text{and } Z$ are continuous random ...
VeniVidiVici47C's user avatar
5 votes
1 answer
136 views

General formula for reversing double integral bounds

The double integral over the region: $$ R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\} $$ is expressed as $$ \...
LightninBolt74's user avatar
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1 answer
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How do I find the volume of this body?

I need help with finding the volume of the body given by $B = \{(x,y,z) \in D \colon 0\leq xyz \leq t^3\}$ Where $D \subset \mathbb{R}$ is given by $0 \leq x \leq t, 0 \leq y \leq t, 0 \leq z \leq t$. ...
nazorated's user avatar
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Integration in high dimensional (n>3) space where integration region is defined by multiple hyperplanes

I'm new to applying integration in high dimension especially larger than 3. Being quite desperate, I searched a lot about it but could not find the clear answer. The problem is, I want to integrate a ...
Happy Jung's user avatar
1 vote
1 answer
59 views

Integrate a sum of trig function under absolute value

Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral: $$ \operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{\,\,n}}\, \left\vert\rule{0pt}{4mm}\,{\cos\...
MathRevenge's user avatar
1 vote
0 answers
49 views

Integrating the multivariate normal distribution over an ellipse

I have learned that if we have two real-valued random variables $X$ and $Y$ that follow the centered multivariate normal distribution, this means that there exists a $2 \times 2$ symmetric matrix $A$ ...
Polyjuice Potion's user avatar
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2 answers
61 views

How to calculate a multiple integral over a triangular region

I am having trouble computing the multiple integral \begin{equation} \int_0^1 \int_0^{1-x} e^{\frac{1}{2}(x + y)^2} \, dy\, dx \end{equation} Because integrating $e^{\frac{1}{2}(x + y)^2}$ with ...
Christopher Miller's user avatar
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0 answers
45 views

Calculating the volume of a body given by $g(x,y,z) = xyz$ using triple integrals

I need help calculating the volume of the region/body/solid given by $D = \{(x,y,z) \colon 0 \leq xyz \leq 8 \text{ and } 0 \leq x \leq 2, 0 \leq y \leq 2, 0 \leq z \leq 2\}$, I am supposed to do it ...
nazorated's user avatar
0 votes
1 answer
44 views

Double Integral Problem in Polar Coordinates

How do I solve this integral? I have tried adding and subtracting $\left(R \cos \phi\right)$ so that I can take denominator as the new term and use it in the iterated integral of $r$, splitting the ...
Anonymousstriker38596's user avatar
1 vote
1 answer
62 views

$\iiint_{\mathbb{R}^3} e^{-\max \left(|x|^3,|y|^3,|z|^3\right)} x^2 y^2 z^2 d x d y d z$

Problem: Calculate $I = \iiint_{\mathbb{R}^3} e^{-\max \left(|x|^3,|y|^3,|z|^3\right)} x^2 y^2 z^2 d x d y d z$ Attempt: denote $ f(x,y,z) = e^{-\max \left(|x|^3,|y|^3,|z|^3\right)} x^2 y^2 z^2 $, ...
hazelnut_116's user avatar
  • 1,709
1 vote
1 answer
146 views

Triple Schwinger integral

I'm working on a perturbative QFT problem which requires three Schwinger parametrizations; that is, $$ \frac{1}{A^n}=\frac{1}{\Gamma(n)}\int_0^\infty dz\ e^{-A z}z^{n-1}. $$ In the end, after taking ...
y9QQ's user avatar
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4 votes
0 answers
53 views

evaluate the volume of solid

Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$. Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
Student's user avatar
  • 319
2 votes
1 answer
151 views

Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$

Problem: Assume $ a^2 + b^2 + c^2 = 1 $. Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$ where $B(\mathbf{0}, 1)=\left\{x^2+y^...
hazelnut_116's user avatar
  • 1,709
2 votes
0 answers
63 views

Calculate $\int_{\substack{x_1\geq 0,\ldots, x_n \geq 0\\ x_1+\cdots+x_n \leq 2}}\sqrt[n]{x_1\cdots x_n\left(x_1+\cdots+x_n\right)} dx_1\cdots d x_n$

Calculate: $$I = \int_{\substack{x_1 \geq 0, \ldots, x_n \geq 0 \\ x_1+\cdots+x_n \leq 2}} \sqrt[n]{x_1 \cdots x_n\left(x_1+\cdots+x_n\right)}\, d x_1 \cdots d x_n$$ Attempt: Perform the ...
hazelnut_116's user avatar
  • 1,709
0 votes
2 answers
39 views

set the limits of integration of the spherical coordinates between two paraboloids and a plane

Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
Student's user avatar
  • 319
0 votes
1 answer
61 views

Calculate the volume of solid

Calculate the volume of solid consisting of the cylinder $x^2+y^2\leq 4, 0 \leq z \leq 2$ and by cone $x^2+y^2\leq z^2, 2\leq z \leq 5.$ I tried to draw the figure on geogebra and I'm trying to use ...
Kalashinikov's user avatar
0 votes
1 answer
98 views

Having a hard time solving this integral $\int_{0}^{1}(\int_{x^2}^{1} 4xe^{y^2} dy)dx$

I am doing some practice exams for multivariable calculus and I am having some trouble solving the follwing integral. $$\int_{0}^{1}\left(\int_{x^2}^{1} 4xe^{y^2} dy\right)dx$$ I know that the ...
Minimo's user avatar
  • 43
1 vote
0 answers
25 views

Double Integral of Bivariate Gaussian Distribution PDF on a Rotated Square

It is well-known that taking the double integral of a Bivariate Gaussian Distribution across a rectangle aligned with the axes is very trivial. However, I am currently considering this double integral ...
Shiran Yuan's user avatar
0 votes
1 answer
44 views

double integral convert from cartesian to polar [closed]

I have this question but I'm stuck on the integral - usually the r gets cancelled out but this time they're both r^2? I can't tell what I'm not seeing or doing wrong.
reluctant programmer's user avatar
1 vote
0 answers
213 views

Inverse square law-type integral over two line segments

This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in ...
Mel's user avatar
  • 156
5 votes
2 answers
220 views

Evaluate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x \text{d}y\text{d}z$

How to evaluate $$ I=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x \text{d}y\text{d}z? $$ Some simple calculation shows that $$ I=\frac{\sqrt{2} -1}{4}\pi+\frac{\pi^...
Setness Ramesory's user avatar
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0 answers
35 views

A 2N variable integral with a delta dirac condition of the inverse functions

I have to calculate the following integral: $$ \int_{x_1=-1}^{1} \int_{y_1=-1}^{1} \cdots \int_{x_N=-1}^{1} \int_{y_N=-1}^{1}\left( \prod_{i=1}^{N} \prod_{j =1, \neq i}^N \frac{1}{|\vec{r_i}-\vec{r_j}|...
javad bashiri's user avatar
0 votes
1 answer
89 views

Integral of $\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right)$

$\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right) =0$ where $r$ is in spherical coordinates and represents the distance from the origin. In the Griffith' Electrodynamics pg 46, first line it is stated ...
Samar Sidhu's user avatar
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0 answers
30 views

Calculate probability of Gaussian random variable exceeding threshold when sampled from different Gaussian distribution

I am wondering if there is a closed-form expression for the probability that a Gaussian random vector $\boldsymbol{X}$ falls in-between some bounds as specified by a different Gaussian random variable ...
Bart Wolleswinkel's user avatar
4 votes
2 answers
215 views

Integrate $\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$

I was trying to solve this integral while solving an electromagnetics problem in physics. $$\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$$ My approach My idea was to ...
sunghoon kim's user avatar
2 votes
1 answer
74 views

Understanding limits of integration after transformation $(x,y) \mapsto (x-y,x+y)$

Consider the following double-integral: $$ \iint_{[0,a]\times[0,a]} (x-y)^2 dxdy \stackrel{*}{=} \int_0^a \int_0^a (x-y)^2 dxdy = \frac{a^4}{6} $$ where $*$ follows by iterated integration. Suppose ...
WeakLearner's user avatar
  • 6,096
1 vote
1 answer
80 views

Evaluating triple integral under linear transformation

Given linear transformation: $$T: \mathbb R^3 \to \mathbb R^3, T(x,y,x)= (3y+4z, 2x-3z, x+3y) $$ We need to evaluate the triple integral : $$\int \int \int_ {T(C)} (2x+y-2z) dx dy dz $$ where $C= \{{(...
S.S's user avatar
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