Questions tagged [multiple-integral]
For questions regarding computation and results related to integrals in at least 2 variables.
1,799 questions
2
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Double Integral of $3y$ over a Region Defined by a Circle
I am trying to evaluate the following double integral:
$$ \int_0^1 \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} 3y \, dx \, dy $$
Can anyone help me with the solution or provide insights into the process?
...
- 21
3
votes
0
answers
72
views
How to show integral of $\frac{f(1,2)}{1+3u^2}$ is equal to integral of $\frac{f(2,1)}{1+3u^2}$?
$\displaystyle \text{... where }f(a,b) = \frac1{1+as^2+bu^2}\text{, and }[s,u]\in[0,1]\times[0,\infty)$.
According to this post $(1)$ and this post $(2)$,
$$\begin{align*}
\int_0^\tfrac1{\sqrt2} \frac{...
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votes
0
answers
64
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Bizzare definite integral
Can anyone help me with the improper integral below:
$$
I = \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} \frac{\sin\left(\sqrt{x^2 + y^2 + z^2 + 1}\right)}{\sqrt{x^2 + y^2 + z^2 + 1}} \cos(x) ...
- 21
7
votes
1
answer
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Several variations of Ahmed integrals: $\int_0^1\frac1{(1+y^2)\sqrt{2+y^2}}\arcsin\left(\sqrt{\frac{2-y^2}{4}}\right )\text{d}y=\frac{11\pi^2}{288}.$
The following are four integrals resembling Ahmed integrals:
$$
\begin{aligned}
&I_1=\int_{0}^{1}
\frac{1}{\left ( 1+y^2 \right )\sqrt{2+y^2} }
\arcsin\left ( \sqrt{\frac{2-y^2}{4} } \right )
...
- 5,840
1
vote
0
answers
32
views
Solving for Hamming Weight Inequality in $\mathbb{F}_2^n$
Given random value vectors $X_i \in \mathbb{F}_2^n$, for $i = 1, 2, \ldots, m$, and a target range $[a, b]$, the objective is to efficiently find all solutions to the inequality $a \leq w_H\left(\sum_{...
- 11
1
vote
1
answer
82
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Evaluating $\iint_R ye^x \,dA$ where $R=\{(x,y)|x^2 + y^2 \le 25, x\ge0, y\ge0\}$
Problem: Evaluate
$$I =
\iint_R ye^x \,dA $$ where $R=\{(x,y)|x^2 + y^2 \le 25, x\ge0, y\ge0\}$
My attempt: The limits are easy to calculate. We integrate with respect to $r$ from $0$ to $5$ and $\...
- 31
5
votes
1
answer
206
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Generalizing $\int_{0}^{\infty} \frac{1}{( 1+x^2) ( 1+(2+x)^2) }\text{d}x+\int_{0}^{1}\frac1{(1+x^2)( 1+(2-x)^2)}\text{d}x=\frac{\pi}{8} .$
It's easily seen that
$$
\int_{0}^{\infty}
\frac{1}{\left ( 1+x^2 \right )\left ( 1+(2+x)^2\right ) }
\text{d}x+
\int_{0}^{1}
\frac{1}{\left ( 1+x^2 \right )\left ( 1+(2-x)^2\right ) }
\text{d}x=...
- 5,840
1
vote
1
answer
63
views
Trouble with reduction of a double integral
I want to compute
$$
I = \iint_{C}\frac{x}{1 + {\rm e}^{y}}\,
{\rm d}x\,{\rm d}y
$$
where $C= \left\{(x, y)\in \mathbb{R^{2}} : x\leq 4,\ y\leq 0,\ y + \ln\left(x/2\right) \geq 0\right\}$
I thought ...
- 644
1
vote
0
answers
84
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Evaluating $\int_0^1\int_0^{1-x}\cos((x+y)^2)\;dydx$
Evaluate:
$$\int_0^1\int_0^{1-x}\cos((x+y)^2)\;dydx$$
This is a problem from a book that I'm reading. The provided solution used the change of the variables $u=x+y$ and $v=y$. I think it is easy to ...
- 861
0
votes
0
answers
36
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Integral limits for $u=y-x$ and $v=y+x$ in $\int_0^2\int_0^{2-x}e^{\tfrac{y-x}{y+x}}dydx$
In order to evaluate the following integral, I used the change of variables $u=y-x$ and $v=y+x$.
$$\int_0^2\int_0^{2-x}e^{\tfrac{y-x}{y+x}}dydx$$
We have $0\le x\le2$ and $0\le y\le 2-x$. Using $y=\...
- 861
1
vote
1
answer
60
views
How to recognize change of variables $x=u\sec v$ and $y=u\tan v$ in $\iint_Re^{x^2-y^2}dA$, where $R$ is enclosed by $x^2-y^2=1, x^2-y^2=16, x=2y$?
Here is a problem from my book:
Evaluate $\iint_Re^{x^2-y^2}dA$, where $R$ is a region in the first quadrant, enclosed by the curves $x^2-y^2=1$, $x^2-y^2=16$, and $x=2y$. (Hint: use the following ...
- 861
1
vote
1
answer
68
views
$\iint_Re^{x^2-y^2}dA$, where $R$ is enclosed by $x^2-y^2=1$, $x^2-y^2=16$, and $x=2y$.
Evaluate $\iint_Re^{x^2-y^2}dA$, where $R$ is a region in the first quadrant, enclosed by the curves $x^2-y^2=1$, $x^2-y^2=16$, and $x=2y$.
I tried change of variables using Jacobian: Let $u=x^2-y^2$ ...
- 861
0
votes
1
answer
28
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Integral against radially symmetric function: can I use some sort of "evenness"?
Suppose $\Omega = [-L,L]^{d} \subset \mathbb{R}^{d}$ and let $f$ be a real-valued and radially symmetric function on $\Omega$. Suppose I have $d$ functions $g_{1},...,g_{d}: [-L,L] \to \mathbb{R}$ and ...
- 987
1
vote
0
answers
51
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Change integration order $\int_{-2}^2\mathrm dx\int_{-2+\sqrt{4-x^2}}^{2+\sqrt{4-x^2}}f(x,y)\mathrm dy$
How do I change the order of this integral?
$$\int_{-2}^2\mathrm dx\int_{-2+\sqrt{4-x^2}}^{2+\sqrt{4-x^2}}f(x,y)\mathrm dy$$
I tried to plot a graph of this function and got two semicircles with ...
- 11
0
votes
0
answers
83
views
Closed-form for $\int_0^1\int_0^1\int_0^1\sqrt{x^2+y^2+z^2}dxdydz$ [duplicate]
RMM mathematical magazine question:
$$I=\int_0^1\int_0^1\int_0^1\sqrt{x^2+y^2+z^2}dxdydz=?$$
By symmetry
$$I=\frac18\int_{-1}^1\int_{-1}^1\int_{-1}^1\sqrt{x^2+y^2+z^2}dxdydz$$
Let us consider the ...
- 14k
1
vote
0
answers
50
views
Compute the integral of exp function over a multi-dimension space
Let $a, z\in\mathbb{R}^d$. Let $Z:=\{z: -1\leq a^{\top}z\leq 0, \|z\|_{\infty}\leq 10\}$.
Can we compute
$$
\int_{Z}\exp(a^{\top}z) dz
$$
by using some known algorithms? Or this problem is NP-hard?
...
- 305
1
vote
2
answers
58
views
Linear substitution in a double integral
Evaluate the following integral, using appropriate linear substitutions: $$V=\int \int_R 2x^2+4xy+3y^2$$ where $R$ is the region enclosed by $y=-\dfrac{3}{2}x+2, y=-\dfrac{3}{2}x+4, y=-\dfrac{1}{4}x,y=...
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2
votes
2
answers
94
views
Solving $\int_{-a}^{a}\int_{-a}^{a}\frac{1}{\sqrt{x^2+y^2}}\mathrm{d}x\mathrm{d}y$ through substitution.
I'm trying to solve the double integral
$$\int_{-a}^{a}\int_{-a}^{a}\frac{1}{\sqrt{x^2+y^2}}\mathrm{d}x\mathrm{d}y$$
by substitution, i.e. $x = r \cos(\theta)$ and $y = r \sin(\theta)$.
The intrgral ...
- 31
1
vote
0
answers
115
views
An integral equation in a Jordan measurable area
We asuume that $f(x,y)$ is defined on $D\backslash{(0,0)}$,within
$$D=\{(x,y)\in R^2| 0\leq x\leq 1, 0\leq y\leq 1 \}.$$
For any open Jordan measurable area $\Delta$ including $(0,0)$, $f(x,y)$ is ...
- 11
0
votes
0
answers
32
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Work verification: Integral of $f(x, y, z) = x^3 + y^3 + z^3$ in the region bounded by the sphere $(x-a)^2 + (y-a)^2 + (z-a)^2 = a^2$
I have to compute the integral $\iiint_V (x^3 + y^3 + z^3) \, dx \, dy \, dz,$ where $V$ is the region bounded by the shpere $(x-a)^2 + (y-a)^2 + (z-a)^2 = a^2$ ($a>0$), that is, $V = \{ (x-a)^2 + (...
- 363
-1
votes
1
answer
84
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Integrating $f(x,y)=x^2+y^2$ over part of circle
I'm trying to solve the following problem:
If $f(x,y)=x^2+y^2$ represents the population density of a planar region on the Earth, where $x$ and $y$ are measured in miles, find the number of people in ...
- 11
3
votes
2
answers
73
views
Missing moment of inertia of solid sphere by a factor of $2$
The moment of inertia of a solid sphere is given by $\dfrac{2MR^2}{5}$.
(image credits)
Here is the process,
$$\mathrm dm =\frac{M}{V}r^2 \sin(\theta) \, \mathrm dr \, \mathrm d\phi \, \mathrm d\...
- 55
1
vote
0
answers
155
views
Change of variables along vertical strip
I'm working on an integral involving multiple Hankel transforms. After the integrating over the Bessel functions, I'm left with a double integral over the region $\mathcal{R}= \{x,y:|x-y|<1<x+y\}...
- 59
1
vote
2
answers
65
views
Evaluate $\int_0^4 \int_{x^{1/2}}^2\frac1{4+y^3}dydx$ [closed]
Please help me to evaluate the following double integral:
$$\int_0^4 \int_{x^{1/2}}^2\frac1{4+y^3}dydx.$$
I can't seem to get an answer that makes sense but online calculators can find a solution so I ...
2
votes
1
answer
93
views
Double Integral Simplification Question
I was reading Brauer and Nohel's ODEs book and came upon the following equality on page 110:
$$
\int \limits_{0}^{t} \int \limits_{0}^{s} g(\tau, \phi(\tau)) d \tau ds = \int \limits_{0}^{t} \left( \...
- 133
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votes
0
answers
16
views
Riemann integrability of the composition of an integrable multivariate function and a $C^1$ homeomorphism mapping
$f(\boldsymbol{x})$ is Riemann integrable on a volume-measurable bounded closed domain $\Omega\subset\mathbb{R}^n$, and $\boldsymbol{u}(\boldsymbol{x})$ is a $C^1$ homeomorphism mapping from a volume-...
- 148
2
votes
1
answer
66
views
Volume inside a cardioid-based cylinder and a sphere
I'm trying to compute the volume inside both the sphere $x^2 + y^2 + z^2 = a^2$ and the cylinder whose generatrices are parallel to the $Z$ axis and whose intersection with the plane $z = 0$ is the ...
- 363
3
votes
2
answers
63
views
Calculating the Volume Enclosed by a Symmetric Surface
Problem. Calculate the volume enclosed by the surface:
$$ z^2 = (x^2 + y^2)(1 - x^2 - y^2) $$
Attempt. I have attempted to transform the equation into cylindrical coordinates. Using the ...
- 337
7
votes
1
answer
179
views
Double integral $\int\limits_{y = 1}^\infty {\int\limits_{x = 1}^y {\frac{1}{{xy{{\left( {y - x} \right)}^{\frac{1}{5}}}}}dxdy} } $ [closed]
I am trying to evaluate this integral
$$I=\displaystyle\int\limits_{y = 1}^\infty {\int\limits_{x = 1}^y {\frac{1}{{xy{{\left( {y - x} \right)}^{\frac{1}{5}}}}}dxdy} }$$
By using CAS checks, the ...
- 2,887
1
vote
0
answers
57
views
Radon transform over a rhombus
My following question is related to an integral over a rhombus $|x|+|y| \leq 1$
$$
I = \int^1_{-1}dy\int_{-(1-|y|)}^{1-|y|}dx\ \delta(r-y-x z)\sum_{n,m}p_{mn}x^m y^n\,,
$$
where $m\in\{0,2,4,...\}$, $...
- 559
0
votes
1
answer
36
views
How to find the bounds of integration from the equations of the surfaces that enclose a volume of triple integral and solve it? [duplicate]
I'm supposed to find the volume enclosed between the two surfaces $x^2+y^2+z^2=4$ and $x^2+y^2=3z$,
when I did it in cartesian coordinates I got some bounds but the integral became a mess.
Here are ...
- 205
0
votes
2
answers
78
views
Letter under double and triple integrals [closed]
There are several letters that can be under a double integral, like $S$, $R$, or $C$ for line integrals. What do all of these letters stand for?
$$
\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \...
- 35
0
votes
1
answer
59
views
Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
- 380
0
votes
2
answers
52
views
Stokes’ theorem question for cylinder-like surface with two disjoint boundaries
Q: Consider $S=\{(x,y,z)\mid x^2+y^2+z^2=25,\; 0≤z≤4\}$ and the vector field $\vec F(x,y,z)=(y,-z,x)$. Use Stoke’s theorem to calculate $\iint_S(\nabla \times \vec F )\cdot \vec n \; dS$
$\vec n$ is ...
- 629
-3
votes
2
answers
174
views
Theorem 16.5, Munkres' Analysis on Manifolds [closed]
In Munkres' Analysis on Manifolds, page 142 Theorem 16.5 it states:
$$\int_{D}f\leq\int_{A}f$$
at the end of that page.
Here, $D=S_{1}\cup\cdots\cup S_{N}$ is compact since $S_{i}=Support\phi_{i}$ and ...
- 380
1
vote
0
answers
42
views
Parametric Curve
Evaluate the following double integral: $I=\iint_{D}^{}y dydx$ with the region $D$ defined by $\left\{\begin{matrix}
x=R(t-sint) & & \\
y=R(1-cost), y=0 & & \\
0\leq t\leq 2\pi&...
- 41
2
votes
2
answers
97
views
Why this simple volume problem in Multivariate Calculus seems to have an anomaly?
Find the volume of the solid inside the cylinder $x^2+y^2-2ay = 0$ and between the plane $z = 0$ and the cone $x^2+y^2 = z^2$.
I tried solving this problem as follows:
Equation of the cylinder $x^2+(y-...
- 3,450
1
vote
0
answers
69
views
A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
- 3,450
1
vote
0
answers
24
views
Computing the multiple-integral of a $n$ variable symmetric function
I'm interested in evaluating the following integral analytically, numerically, or using a combination of both. Let
$$ I_n(\rho) = 2^n\int_{\mathbb{R}_+^n} \sqrt{1+ \rho^2 \left(\prod_{k=1}^n x_k^{-2}\...
- 1,095
0
votes
1
answer
76
views
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 ...
- 5,480
0
votes
0
answers
58
views
Integral of two-variable function over level sets
Suppose I wish to compute the following integral:
$$
\int_{D}\mathrm{d}x\,\mathrm{d}y\,f(x,y)
$$
where the domain of integration $D$ is given by
$$
D=\left\{(x,y):0\leq g(x,y)\leq 1\right\}
$$
Suppose ...
- 933
0
votes
1
answer
89
views
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 ...
0
votes
0
answers
51
views
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 = ...
- 99
4
votes
1
answer
159
views
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 $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
- 884
1
vote
1
answer
44
views
Probability densities with conditions - how to find the distribution function
I have two probability density functions where i need to find the distribution function.
The first function is
$$f(x,y)=
\begin{cases}
\frac{x}{y} & \text{for $0\leq x\leq y\leq c$}\\
0&\text{...
- 11
0
votes
0
answers
40
views
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 ; ...
- 143
0
votes
1
answer
54
views
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 ...
- 143
2
votes
1
answer
136
views
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 ...
- 55
0
votes
0
answers
20
views
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 $[0,1]$...
- 245
1
vote
1
answer
43
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\...
- 451