Questions tagged [multiple-integral]
For questions regarding computation and results related to integrals in at least 2 variables.
1
vote
2answers
18 views
Evaluating this double integral with polar coordinates
Find $\int_\gamma x^2y^3dx+y\,dy$ where $\gamma$ is the positively oriented border of $\{(x,y) \in R^2 : 1\le x^2+y^2 \le4\} $
If I convert to polar coordinates and sketch the domain, I get that $$\...
0
votes
0answers
16 views
Shapes that maximize gravity at fixed mass and density
Suppose we take the set of all bodies $S\subset\mathbb{R^3}$ for which the limit
$$
\lim_{r\rightarrow\infty}m_J(S\,\cap B(0,r))
$$
exists and is a fixed number. Here, $m_J$ denotes the Jordan content ...
0
votes
1answer
31 views
Find $\iiint_V z$ with $V=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$
Let $f(x,y,z)=z$ and $T=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$
Find $\iiint_T f(x,y,z) dV$
I'm having a few problems with this integral, here's ...
0
votes
1answer
37 views
Flux of vector field across surface via divergence theorem and directly
There is vector field $F = [x,y,-z]$. We need to find the flux of the vector field outward across the given surface $\sigma=x^2+y^2+z^2=1, x\ge0, y\ge0, z\ge0$ directly and by using Gauss theorem (i.e....
1
vote
0answers
23 views
Finding theta when using polar coordinates in solving double and triple integrals
So I have a problem with finding theta. I can always see from where to where the theta is going. The problem occurs when calculating volumes. Sometimes in the solutions book, for the full circle
$$4\...
0
votes
0answers
22 views
Integration by parts, double integral and PDE
I have a function $v=v(x,y,t)$ satisfying the following PDE (degerate heat equation)
$\frac{\partial v}{\partial t} - \Delta_x v - \Delta_y v - 2 \nabla_x \cdot \nabla_y v = 0$.
I also have a ...
1
vote
1answer
57 views
Surface integral of $f(x) = \frac{1}{ \Vert x -x_0 \Vert } $ over sphere
Let $S \subseteq \mathbb{R}^3$ the sphere of radius $r$ centered at the origin. Let $x_0 \in \mathbb{R}^3$ be such that $x_0 \notin S $.
Let $f:S \to \mathbb{R}$ be such that $f(x) = \dfrac{1}{ \Vert ...
1
vote
0answers
20 views
Notation question for push forward substitution in iterated integral
Consider the following iterated integral
\begin{gather*}
\int_{0}^{1}\int_{0}^{\pi}y\sin(xy)d xd y.\tag{1}
\end{gather*}
It is standard to evaluate (1) as follows.
\begin{align*}
\int_{0}^{1}\int_{...
0
votes
1answer
18 views
How to calculate double integral when x and y are not to append to each other
I have a hard time understanding how to integrate a double integral when I have a straight line and function like: $y=\frac{x}{3}$ and $x = y^2$ for the integral $f(x,y) = (x-y^2)$.
I evaluate it ...
1
vote
1answer
43 views
Finding $\iiint 6z\,dx\,dy\,dz$ over $\lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$
I want to calculate integral : $\iiint 6z\,dx\,dy\,dz$. The area is $$\Omega= \lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$$
My problem is this, that I don't know what is ...
1
vote
1answer
38 views
Computing to volume of ball intersection with cone and another ball
I am trying to calculate the integral $\ \int\int\int_V z dV $ where $\ V $ is the volume inside the ball $\ x^2 + y^2 + (z-2)^2 = 4 $ and also inside the cone $\ z^2 = x^2 + y^2 $ and outside the ...
3
votes
4answers
553 views
Computing volume inside a ball and outside a cylinder
I try to calculate the volume inside the ball $\ x^2 + y^2 + z ^2 = 4 $ the outside the cylinder $\ x^2+y^2=2x $ using double integral.
Since the shape is symmetric I chose $\ z = 0 $ as bottom limit ...
2
votes
0answers
23 views
Simplify $\int_{\Omega} \int_{\Omega} 1_{A}(\omega) g(\tilde{\omega}, \Pi(\omega)) d\mathbb{P}(\tilde{\omega}) d\mathbb{P}(\omega) $
Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space and $D$ be a compact topological space, equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(D)$.
Furthermore, ...
0
votes
2answers
38 views
How do I go about choosing the correct bounds for this probability density function?
Given the function $f(x,y) = 24xy \quad$ for $\quad 0 \le x \le 1 \quad$, $\quad 0 \le y \le 1 \quad$, and $\quad 0 \le x+y \le 1 \quad$.
I want to show that the double integral of the function ...
0
votes
0answers
25 views
To calculate volume of a solid via triple integral in spherical/cylinder coordinates
I come across on a task to calculate the volume of the solid bounded by surfaces as below via triple integral.
$x^2+y^2 \le 3z^2$ and $x^2+y^2+z^2 \le 4$
The first surface seems to be the infinite ...
1
vote
1answer
31 views
Multidimensional integral induction
I am trying to integrate the region under $\sum_{i=1}^{n}{x_i}=1$. What I've done is write out the multiple integral:
$$\int_{0}^{1}\int_{0}^{1-x_1}...\int_{0}^{1-\sum_{i=1}^{n-1}{x_i}}1 \,dx_n\,dx_{...
0
votes
1answer
30 views
Calculating the area of the intersection between $S: x^2+y^2+z^2=4$ and $z\ge1$.
I started by drawing both graphs and found that the intersection is just the part of the sphere above $z=1$. So it's the part of the sphere from $1\le z\le2$ and let this be called $S^1$. I then let $...
0
votes
0answers
16 views
Center of the mass of the upper part of an ellipsoid
I want to find a center of the mass of the upper part of a homogenous ellipsoid: ${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2} \le 1$, where $a,b,c > 0$. Because of symmetry, I need to find only ...
1
vote
1answer
46 views
Multiple integrals and uniform continuity
For a function $$f (x,y)$$ continuous on $[a,b]\times J$, where $J$ is an open interval:
$$g(y) = \int_a^b\ f(x,y)\ \mathrm dx$$ is also continuous.
My question is: if $J$ is a closed then $f$ ...
1
vote
0answers
30 views
Compute the volume of $ E(R,r)=\left\{ (x_1,x_2,x_3,x_4)\in\mathbb R^4:\frac{x_1^2+x_2^2+x_3^2}{R^2}+\frac{x_4^2}{r^2}\le 1 \right\}$
For positive real numbers $R$ and $r$, let
$$ E(R,r)=\left\{ (x_1,x_2,x_3,x_4)\in\mathbb R^4:\frac{x_1^2+x_2^2+x_3^2}{R^2}+\frac{x_4^2}{r^2}\le 1 \right\}. $$
Find the volume of $E(R,r)$ by ...
0
votes
1answer
22 views
Approximate area using multiple integral
The area of the surface described by $z = f(x,y)$ for $(x,y)\in R$ is given by
$$\iint_R\sqrt{1 +[f_x(x,y)]^2+[f_y(x,y)]^2}dA$$
Find an approximation to the area of the surface on the hemisphere $x^2+...
0
votes
0answers
8 views
Double integration w.r.t. a transform kernel.
I am trying to work with Linear Canonical Transform. There is particular property, the additivity of transform i need to prove.
Specifically let $$\tilde{f}^M(y)=A_b \int{f(x)\exp^{\frac{j}{2b}(ax^2-...
0
votes
1answer
30 views
Change of variables in triple integral
Let $D$ be the region in $xyz-$space defined by inequalities $1 \le x \le 2, 0 \le xy \le 2 $ and $0\le z \le 1$. I want to evaluate $\displaystyle \int\int\int_D (x^2y + 3xyz) \text{dxdydz}$ by ...
13
votes
3answers
244 views
Find closed form for quadruple integral
I am trying to find a closed form of the following integral
$$
\int _0^{\infty }\int _0^x\int _0^y\int _0^z \exp \left( -\frac{a x^2}{2}-\frac{b y^2}{2}-\frac{c z^2}{2}-\frac{d w^2}{2} \right) \,\...
2
votes
1answer
75 views
Multivariate Kolmogorov distance bounded by Wasserstein distance
I'm trying to find a bound for the multivariate Kolmogorov distance in terms of the Wasserstein distance.
Denoting by $F$ and $G$ two cumulative distribution functions (cdf) on $\mathbb{R}^n$ the ...
0
votes
0answers
35 views
Triple integral to calculate the volume of pyramid vs it's formula
It is well known the formula to calculate the volume of a pyramid:
$V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the base to the apex.
However I need to ...
0
votes
1answer
27 views
Evaluate double integral bounded by lines and hyperbolas
Evaluate the integral $$\iint_R x^2 y^2 dx dy,$$ where $R$ is the bounded portion of the first quadrant bounded by the lines $y=x, y=4x$ and the hyperbolas $xy=1$ and $xy=2.$
Based on the graph of ...
0
votes
1answer
17 views
Evaluate double integral on a circle
Let $f$ be defined on $Q = [ -1,1 ] \times [ -1, 1 ]$ as follows: $f(x,y) = x^2+y^2$ if $x^2 + y^2 \le 1,$ and otherwise, $f(x,y) = 0.$ Compute the volume of this ordinate set.
I tried expressing ...
2
votes
0answers
54 views
A double integral with a closed form, generalization
I have encountered a double integral with three parameters which has the following form:
$$I(a,b,c)= \int_{-\infty}^\infty \int_{-c}^\infty \frac{e^{-a (x^2+b x+y)} \mathrm d y \mathrm d x}{\sqrt{(x^...
1
vote
2answers
61 views
Rudin Principles of Mathematical Analysis Chapter 10, Exercise 8
I'm working on exercises of chapter 10 in Baby Rudin.
I refer to R. Cooke's solutions manual to Baby Rudin while I'm solving those exercises.(https://minds.wisconsin.edu/handle/1793/67009)
But I ...
0
votes
0answers
30 views
Nested integral with same function at every integral being multiplied an integrated again.
$$\int_z^{\infty}f(z_1)\int_{z_1}^\infty f(z_2) \cdots \int_{z_{n-1}}^\infty f(z_n)dz_ndz_{n-1}\cdots dz_2dz_1$$
Just for reference we could also write it as
$$\int_z^\infty \int_{z_1}^\infty\cdots\...
0
votes
0answers
18 views
Volume by rotation using polar coordinates
I have been trying to solve an exercise taken from the legendary brazilian book "Um Curso de Cálculo Volume 3" about change variable in double integral. This exercise literally says:
Consider the ...
0
votes
1answer
45 views
Evaluate the volume bounded by $z=1-x^2-y^2$ and $z=1-y.$
Evaluate the volume of $V\subset \mathbb{R}^3$, which is bounded by paraboloid $z=1-x^2-y^2$ and the surface $z=1-y,$ for $z\geqslant 0.$
Attempt. The desired volume goes like:
$$\iint_D \big(1-x^2-...
0
votes
0answers
4 views
How to carry out differentiation of double integral using Lebinitz theorem?
I have the following integral:
$I=\int_{-h_1}^{-h_2} \int_{t+s}^{t} (h_{12}-h_{21})\dot x^T(\alpha)Y_2 \dot x(\alpha)d\alpha ds$
where $t\in\mathbb{R}_+$, $x\in \mathbb{R}^n$, $\dot x=\frac{dx}{dt}$ ...
3
votes
1answer
67 views
6D Fourier transform of Coulomb potential
I just wanted to check my result.
Let's define the Fourier transform as (the integral over whole real line):
$$g(k)=\frac{1}{\sqrt{2 \pi}} \int e^{-i k x} f(x) dx$$
We have the following function:
...
1
vote
0answers
21 views
Winding number in 4D & SU(2) group
In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
0
votes
0answers
27 views
Uniqueness of solution on Advection Diffusion equation
Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following Advection-diffusion Equation given by
\begin{align}
\nabla\cdot\left(\mathbf{V}\psi-D\nabla \psi\right)&=F\quad \...
0
votes
1answer
17 views
Integrating a Bivariate Piecewise Function
Given $$
f_{ab}(u,v)=
\begin{cases}
(1-a)u^{-a}v,\quad u^a>v^b\\
v^{1-b}, \quad\quad\quad\quad\: u^a<v^b\\
\end{cases}
$$ where the domain is on $(u,v)\in[0,1]\times[0,1].$
How would I split ...
1
vote
1answer
19 views
For Green's theorem, why is the region of integration of the line integral a weird partial derivative character?
Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem?
$$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$
...
1
vote
1answer
62 views
Using transformation to evaluate double integral
Given the transformation $T(x, y) = (x - y, x + y)$, evaluate the double integral
$\iint_R (x^2+y^2) dA$, where $R$ is the rectangle in the $xy$-plane with vertices $A(1, 1)$, $B(2, 2)$, $C(-1, 5)$ ...
0
votes
1answer
38 views
What values of $\rho$ can give the answer? $V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$
Find the Volume of the region bounded above by sphere $x^2 + y^2 + z^2 = 2a^2$, and below
by the paraboloid $az = x^2 + y^2$?
$$V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$...
0
votes
1answer
37 views
Triple Integral in spherical coordinate
$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$;
Hi guys, I don't quite get which region is this, is it that ...
0
votes
0answers
35 views
How can I calculate Fresnel integral via double improper integral?
So, there are many ways to calculate the integral $\int_{-\infty}^{+\infty} \sin(x^2)\,\mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My ...
3
votes
1answer
52 views
How should I solve this triple integral?
Hi everybody I have a triple integral I can't solve:
$$\iiint \sqrt {x^2+y^2+z^2} \,dx \,dy \,dz $$
Which the region is between $z=\sqrt {x^2+y^2}$ and $z=4$ .
The question says after using the ...
0
votes
1answer
65 views
Finding volume of solid under $z = \sqrt{1-x^2-y^2}$ above the region bounded by $x^2 + y^2-y=0$
Find the volume of the solid that is under the hemisphere $z=\sqrt{1-x^2-y^2}$ above the region bounded by the graph of the circle $x^2 + y^2-y=0$.
I solved this problem using limits of integration $...
3
votes
1answer
59 views
Integration with Dirac delta function of two-argument function
I would like to solve that problem:
$$
\int d^2 \mathbf{k} \, \delta(f(k,\phi)) =
\int_{k_1}^{k_2} dk \, k \int_0^{2\pi} d\phi \, \delta(f(k,\phi))
\,,
$$
where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
0
votes
1answer
36 views
Surface area of a sphere over a disc
What's the surface area of the sphere $x^2 + y^2 + z^2 = 1$ over the disc $(x-1/2)^2 + y^2 \le 1/4$ ?
I've tried something, but I don't think it's right, as it's not a "nice answer"
So here is what ...
0
votes
0answers
14 views
Manipulating change of variables formula for multiple integrals with roots
The change of variables for multiple integral can be formulated like:
$\int_D(f \circ g) (x) |det~J_g(x)|dx = \int_{D'}fdx'$
for a continous $f: D'\rightarrow X$ and two-times continous, injective $...
0
votes
1answer
23 views
Changing order of $\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) \,dx\,dz\,dy$ to $\iiint f(x,y,z) \, dy\,dz\,dx$
The sketch of the region is:
Based on it I get:
$$\int_0^{\sqrt{23}}\int_0^{4-y}\int_{\frac{x^2-23}{4}}^{4} f(x,y,z)\, dy\,dz\,dx$$
But this arrangement makes me drag $y$ so I must be missing ...
0
votes
2answers
30 views
Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$
Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$
Not sure how to proceed
