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Questions tagged [change-of-variable]

This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals

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Laplace Transform: Indeterminate Form in Definite Integral Change of Variables Calculation

I was trying to find the Laplace transform of $e^{3t}$: $$\int^\infty_0 e^{3t}e^{-st} \ dt = \int_0^\infty e^{3t - st} \ dt = \lim_{x \to \infty}\int_0^x e^{3t - st} \ dt$$ So if we then attempt to ...
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change of variables with limits

If I have the following function: $$ 1 = \int_0^{R_M}g_s(R)R^{-1}dR$$ where $x_M = R_M/\sigma_0$, and $x = R/\sigma_0$ how would I perform sub. of variables on the limits? I know I would have ...
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change of variable problem

Let $\Delta\geq 1$ be a constant. I need to prove that $$ 2\int_1^T x\int_{T/2x^2}^{2T/x^2}\left(\frac{\sin(\frac{\Delta}{4}\log\frac{2\pi}{t})}{\frac{\Delta}{4}\log\frac{2\pi}{t}}\right)^2 dt\,dx = \...
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Prove that there is a point $c\in(a,b)$ such that $f'(c)=0$.

Let $f:[a,b]\rightarrow\mathbb{R}$, $0<a<b$, a function that is differentiable and bijective such that $\int_{f(a)}^{f(b)}f^{-1}(x)dx=0$. Prove that there is a point $c\in(a,b)$ such that $f'(c)=...
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Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions ...
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16 views

How to do change of variables of a j.p.d.f with N pdf(s)?

Given that I have a joint probability distribution(jpdf) of: $$P(x_1,...,x_N) = C_N \prod_{j=1}^{N}(1-x_j)^a(1+x_j)^b \prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ where $$\prod_{1\leq j <k \leq N} |...
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25 views

Change Variables in a Multiple Integral

Let $D$ be the region that's bound by $y=x^2, y=2x^2, x=y^2, x=3y^2$. $D$ corresponds to the region $E$ where $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$. Sketch $D$ and $E$ Find the solution ...
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16 views

change of variable for integral to calculate posterior distribution

I'm working through an example which can be found here (p. 36), if someone is interested. I have an integral of the form: $$P(x|\mu)=\int d\sigma P(x|\mu, \sigma)P(\sigma)=\int d\sigma \frac{1}{\sqrt{...
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19 views

Change of Variable Bounds of Integration

One of my practice problems asks us to compute the volume of the region enclosed by the unit sphere $\{(x,y,z): x^2+y^2+z^2=1\}$ and the set $\{(x,y,z): z= |x|\}.$ My first intuition is to use ...
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Integration by substitution in $f: \mathbb{R^2} \to \mathbb{R}$ with Jacobian method proof

I've searched everywhere on Google for a somewhat formal proof for the integration by substitution in double integral with the Jacobian determinant method but I can't find any. Anyone knows any ...
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1answer
26 views

Multiple integration with constraints on variables

I have a function $(x_1, x_2)\mapsto g(x_1, x_2)$ where $x_1$ and $x_2$ are both 3D vectors. I would like to integrate function $g$ over the whole space but with some constraints on $x_1$ and $x_2$ ...
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>Calculate the volume of the $n$-dimensional ball using change of variables.

Calculate the volume of the $n$-dimensional ball using change of variables. My attempt. Lemma 1. For each $k \in \mathbb{N}$ we have $$\int_{0}^{\pi}\sin^{k}xdx = \frac{k-1}{k}\int_{0}^{\pi}\sin^...
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About Theorem 6.19 on pp.132-133 in “Principles of Mathematical Analysis” by Walter Rudin

Thank you very much, Saaqib Mahmood, for your text. I copied and pasted it: Theorem 6.19 on pp.132-133: Suppose $\varphi$ is a strictly increasing continuous function that maps an interval $[ ...
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Application of change of variables (volume of balls)

Let $A \subset \mathbb{R}^{n}$ be an open set and $f: A \to \mathbb{R}^{n}$ a $C^1$ function with $Df_{a}$ an isomorphism. Show that for each $a \in A$, $$\lim_{r \to 0}\frac{\mathrm{vol} f(B_{r}(a))...
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Analytic posterior likelihood of bayesian logistic regression with uniform prior

I am trying to show that the posterior of the following model is proper. Assuming, $p(\alpha, \beta) \propto 1$, and $n,x, y$ given. $$ \begin{align} p(\alpha, \beta) & \propto p(\alpha, \beta | ...
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3answers
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Undo this transformation

I have two variables $x$, $y$ and calculate the following: $a = \frac{x}{\sqrt{x^2+y^2}}$, $b = \frac{y}{\sqrt{x^2+y^2}}$ Using $a$ and $b$ is there a way I can derive my original $x$ and $y$?
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Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates

I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ...
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Use the change of variables to determine the density for a uniform distribution on $[a,b]$

Knowing that the density of a uniform random variable on $[0,1]$ is: $f_{U}=\left\{\begin{matrix} 1 & x\in [0,1]\\ 0 & x\notin[0,1] \end{matrix}\right.$ How to determine the density of a ...
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1answer
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How the line element change in a complex change of variables?

So I'm learning conformal field theory and having a hard time to prove the conformal Ward identity. From the lectures notes from John Cardy, he express the integral $$ \delta S = \frac{1}{2\pi} \...
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Define a measure from density function with symmetry

A little bit similar to what I have asked before: Change variable in the integral with nonnegative measure Let $f_X(x)$ be a probability density function with $x\in \mathbb{R}^n$ and $X\subset\...
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1answer
62 views

Integral of $x^n y^m$ on the unit disc

Let $E = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \leq 1 \}$. Compute $$ \int_E x^n y^m \ dx \ dy$$ for all $n,m \in \mathbb{N}$. What is the best way to do this? One can, of course, transform it using ...
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1answer
57 views

$2\int_{0}^{\sqrt{t}}\frac{1}{\sqrt{8\pi}}e^{-\frac{x^2}{8}}dx=\int_{0}^{t}\frac{1}{\sqrt{8\pi}}e^{-\frac{y}{8}}\sqrt{y}dy$?

I encountered the following "claim" in a probability exercise solution (find the distribution of $Y=X^2$ where $X\sim N(0,2)$ $$2\int_{0}^{\sqrt{t}}\frac{1}{\sqrt{8\pi}}e^{-\frac{x^2}{8}}dx=\int_{0}^{...
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1answer
89 views

Spivak, Calculus on Manifolds 3-40

Looking for a hint to the following question: If $g: \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and $\det g'(x) \neq 0,$ prove that in some open set containing $x$ we can write $...
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In search of a trick for a multiple integral with difficult integration limits

I'm trying to make a small model for the expected life-time of some molecules (I'll edit the question and add info if someone wants to know the context) and I reached the following multiple integral: $...
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Change of variables: Apply $\tanh$ to the Gaussian samples

In the paper "Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor" Appendix C, it mentioned that applying $\tanh$ to the Gaussian sample gives us the ...
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Change variable in the integral with nonnegative measure

Just ask a very fundamental problem of changing variable when doing integration. I am a bit confused about the following: Suppose I want to do the integral $$\int_X v(x) d\mu(\gamma^{-1}x)$$ where $...
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1answer
31 views

Change of variables for an integral

I have an integral where I need to change variables. The integral has the form, $\int_0^x f(x,t) dt$ . I change variables/rescale by setting $\tilde{t}=xt$, which means $d\tilde{t}=xdt$. Would the ...
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Rudin's proof of change of variables.

When I read Rudin's proof of change of variables, I have a problem underlined in red below: I don't understand why (31) is true when $T$ is a primitive $C'$-mapping. I know that a primitive mapping ...
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1answer
46 views

Show that $(X,Y)$ has bivariate normal distribution when $X = Z_{1}$ and $Y = Z_{1} + Z_{2}$, where $Z_{i}\sim\mathcal{N}(0,1)$

Assume that $Z_{1}$ and $Z_{2}$ are independent standard normally distributed random variables. Show that $(X,Y)$ has bivariate normal distribution when $X = Z_{1}$ and $Y = Z_{1} + Z_{2}$. MY ...
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1answer
28 views

Transformation of continuous random variables: solution verification

Assume $X$ to be a random variable whose probability density function is given by \begin{align*} f_{X}(x) = \begin{cases} \displaystyle\frac{3x^{2}}{2} & \text{if}\,\,\,-1\leq x \leq 1\\\\ 0 &...
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1answer
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Substitution in complex integral and the Argument Principle.

Let's say $C$ is a simple closed curve in the complex plane and $f(z)$ is holomorphic and doesn't vanish on $C$. According to wikipedia, one can make the following change of variables: $$\omega = f\...
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change of variables in integral - how do limits change?

If I have this integral: $$\int_0^{\sigma_0}xR^2dR$$ and I know that $x=\frac{R} {\sigma_0}$ and I substitute: $$\int x (x\sigma_0)^2 dx= \int x^3 \sigma_0^2 dx$$ what are the new integration ...
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Transformation of random variables: what went wrong?

Assume that $X_{1}$ and $X_{2}$ are independent exponential random variables with parameter $\lambda$. Let $Y_{1} = X_{1} + X_{2}$ and $Y_{2} = X_{1} - X_{2}$. Determine (a) Find the joint ...
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Assume that $X$ and $Y$ have joint probability density function $f_{X,Y}$. Calculate the joint probability density function of $U = XY$ and $V = X/Y$

Assume that $X$ and $Y$ have the following joint probability density function $$f_{X,Y}(x,y) = \begin{cases} \displaystyle\frac{1}{x^{2}y^{2}} & \text{if}\,\,x\geq 1\,\,\text{and}\,\,y\geq 1\\\\ \,...
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Compute probability $P(U_1 \leq a\text{cos}(U_2))$

Consider the independent random variables $U_1 =U(0,c)$ $U_2=U(0,\pi/2)$ The challenge is to compute the probability: $$P(U_1 \leq a\text{cos}(U_2))$$ An attempt: $$P(U_1 \leq a\text{cos}(U_2))=P(...
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66 views

How is this integral computed??

I'm reading this book about electrical properties of materials where the electron is introduced as a wave. Using the equation of a wave, they bring about the "envelope" of a wave. So here is how the ...
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An affine transformation, and its effect on a curve and a polynomial.

Suppose $(u,v) = A(x,y)$ is affine transformation. Where $u = ax + by + e$, and $v = cx + dy + f$ , and the inverse transformation given by $x = a'u + b'v + e'$ and $y = c'u + d'v + f'$. Suppose ...
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Change of variables and the partial derivative

From time to time, I suddenly get confused with a change of variables in a partial derivative. Here, I am trying to perform a change of variables $(x,t) \mapsto (\xi, \eta)$ where $$\xi = t \qquad \...
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1answer
26 views

Formula to calculate change in distance to destination or origin of a straight-line path of travel

I am writing an application that consumes GPS data - and I am trying to calculate direction traveled based on a change in distance to the destination and origin. Assume that I have a straight path of ...
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2answers
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Changing order of integration:$\int_0^\infty\int_{-\infty}^{-y}f(x)\mathrm dx\mathrm dy\Rightarrow\int_{-\infty}^0\int_0^{-x}f(x)\mathrm dy\mathrm dx$

Why does $$\int_{0}^{\infty} \int_{-\infty}^{-y} f(x)\mathrm dx \mathrm dy \Rightarrow \int_{-\infty}^{0} \int_{0}^{-x} f(x) \mathrm dy \mathrm dx$$ The title is pretty self explanatory. I couldn't ...
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1answer
28 views

For which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges?

Im looking for which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges. What I was looking for is an appropriate change ...
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Change of variables in integral: the limits of $y = \int_{\sigma_0}^{R_M} \frac{dR}{R} $

I have this equation: $$y = \int_{\sigma_0}^{R_M} \frac{dR}{R} $$ where $x = R/\sigma_0$ If I want to do a change of variables, I would have: $$y = \int \frac{\sigma_0}{x\sigma_0} $$ $$y = \int \...
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Well-definedness of a fundamental solution by proofing that a set has measure zero with Cavalieri's principle

I am currently walking my way through a paper from the year 2009, where the Malgrange-Ehrenpreis Theorem ("Every linear partial differential operator with constant coefficients has a fundamental ...
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2answers
80 views

Changing variable - Integration goes wrong.

I was trying to do the integration $$I=\frac{\pi}{2}\int_0^\pi \frac{dx}{a^2\cos^2x + b^2\sin^2x}$$ If I divide throughout by $\cos^2x$ and use substitution ($t=\tan x$), I obtain$$I=\frac{\pi}{2}\...
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1answer
33 views

Using change of variables to transform density functions

I'm was working on some exercises on statistical inference and came across a question I could not solve. After a while I decided to take a look at the solution to hopefully understand the problem ...
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1answer
65 views

$\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}$

I'm trying to calculate the integral $$\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}.$$ I've tried in two methods: Regular spherical coordinates, but this leads to really unfun ...
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1answer
28 views

Change of variable in Definite Integrals : Shift by value of Integral [closed]

For an integral of the form(given below) which does not have an anti-derivative $I = \int_{t=0}^\infty f(t)dt$ I wish to bound the value of the integral, by using a bound given in the problem ...
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1answer
84 views

How can this multiple integral be evaluated?

I am stuck trying to solve the following integral: $$\int_R (y+2x^2)(y-x^2) dA$$ where $R$ is defined by the following equations: $xy=1$, $xy=2$, $y=x^2$, $y=x^2-1$ with $x$ and $y$ positives. I've ...
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44 views

Change of variable in 2D integral

If I have an integral of the form $$ I = \int_{-T}^T dx\int_{-T}^T dy\ i(x - y) \tag1$$ Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I ...
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4answers
62 views

Use the spherical coordinates to compute the integral $\int\limits_{B} z^2 dx dy dz$ where B is defined by $1\leq x^2 + y^2 + z^2 \leq 4$

, however the answer I got to is different than the answer sheet. The answer sheet says that it should be $\frac{62}{15}$ Am I making some mistake or is the answer sheet incorrect?