Questions tagged [iterated-integrals]

This tag is for questions relating to iterated integrals. In calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example, $~f(x,y)~$ or $~f(x,y,z)~$) in a way that each of the integrals considers some of the variables as given constants.

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Is the tower property true conditional on vector?

Let $X|\theta \sim F$ with $\theta$ a random vector and $X$ a random variable. Is it true that $$ E[X] = E[E[X|\theta]] $$ even though $\theta$ is a vector? If so, can one say that if $X|\mu,\sigma \...
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Iterated conditional probability notation

I'm currently self-studying Andrew Gelman's book "Bayesian Data Analysis" third edition. At the page 41, they write: $E(\tilde{y}|y)=E(E(\tilde{y}|\theta,y)|y)$ I am ok with multiple ...
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Confusion over changing the bounds of an iterated integral

I have the following: The sum of the iterated integrals $$ \int_{-2}^{-1}\int_{-\sqrt{y+2}}^{\sqrt{y+2}} f(x,y)\, dx\, dy + \int_{-1}^{2}\int_{y}^{\sqrt{y+2}} f(x,y)\, dx\, dy$$ is equal to: a) $\...
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Volumen of the intersection of a cone and a tangent cone.

the professor of our integration class gave us the following exercise as an assignment: Determine the measure in $\mathbb{R}^3$ of $A=\{(x,y,z) \in \mathbb{R}^3 ,x^2+y^2+z^2-2x+2z\leq0\leq x^2-y^2-z^2 ...
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A polynomial sequence by iterated integration

I have been inspired by this question to think about the polynomial sequence $a : \mathbb{N} \rightarrow \mathbb{Q}[x]$ defined by $$a_n(x) = \int_0^{1{-}x}dy\,a_{n{-}1}(y),\qquad a_0 = 1.$$ The first ...
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area of one rose petal using iterated integral but the outer integral is of radius

I have a Rose leaf, described by the equation r=a*sin(3θ), from θ=0 to $\frac{\pi}{6}$. I need to make an iterated integral of this area. It's easy when the outer ...
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Using iterated integrals to get the area with strict conditions

We were tasked to use iterated integrals over this area shaded in yellow here: I'm confused as to how to approach this as I've only seen ones that encompass the whole space inside the closed shape ...
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How to calculate this iterated integral:$\int_0^1dy\int_0^y dx\int_0^x \frac{e^z}{1-z}dz$?

Is there any trivial way to calculate this iterated integral,$\int_0^1dy\int_0^ydx\int_0^x\frac{e^z}{1-z}dz$? I've already solved this by taking series expansion of $e^x$ into the integral,but I ...
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Writing the iterated expectation with a single integral

I would like to write the expected value of $c(x)$ where $x$ is sampled from a distribution $\gamma(x|m)$ and $m$ is sampled from another distribution $\omega(m)$. Here, for any fixed $m$, the ...
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How we solve iterated integrals such as this one?

Here is the problem $(\textbf{20}\text{ points})$ Calculate the following iterated intergals: $$\text{a. } \int_0^2\mathrm dx \int_{-1}^1\big(3x^2-(x+y)e^y+xy^3\big)\ \mathrm dy,$$ I can do the ...
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$\int_a^x \int_a^t f(t_1) dt_1 dt = \int_a^x (x-t) f(t) dt$?

A quick question on confusion on some iterated integral in the text I am studying : Defined in Special Functions by Roy : How $\int_a^x \int_a^t f(t_1) dt_1 dt = \int_a^x (x-t) f(t) dt$ is true? ...
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Ramanujan: If $\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx$, then $\frac\pi2\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty\psi(q,x)\psi(p,nx)dx$.

Apparently, the following appeared in Ramanujan's first letter to GH Hardy. If $$\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx,$$ then $$\frac{\pi}{2}\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty \psi(q,x)\psi(...
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Can we write the right hand side as a one integral formula? (A lecture on multivariable mathematics by Theodore Shifrin on YouTube)

I am watching a lecture on multivariable mathematics by Theodore Shifrin on YouTube. I want to know the answer to this problem. My answer is the following. My answer uses two integral formulae. $$\...
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Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates.

Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates. I could say that: $0\leq 1, 0\leq x\leq z, 0\leq y \leq \...
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Calcuate the integral $\int_1^{-1}dx\int_x^{2x}e^{x+y}dy$

Calcuate the integral $\int_1^{-1}dx\int_x^{2x}e^{x+y}dy$ Here is my working out: Split the integral into their respective derivate parts: - $$\int_1^{-1}e^xdx\int_x^{2x}e^ydy$$ The RHS is given as : $...
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Iterated integrals, range is given by inequalities

This problem is from mathematical statistics 8ed by hogg Let $C = \mathbb R^n$ For $A$ in $C$ define the set function $$ Q(A) = \idotsint\limits_A dx_1 dx_2 \cdots dx_n$$ If $B = \{(x_1, x_2, \dots, ...
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2 answers
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Changing integration order in an iterated double integral

I am trying to solve the integral below (equation 1) by changing the integration order from dydx to dxdy: $$ I = \int_{\frac{2}{\sqrt{5}}}^1\int_{\sqrt{1-x^2}}^{\frac{x}{2}}f(x,y) \mathrm dy \mathrm ...
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Iterative integration with indicator function

I have an exercice in which the result $$ \int_{[0,1]^n}\mathbf{1}_{\{0<x_n<\ldots<x_2<x_1<1\}}\prod_{k=1}^{n}\mathrm{d}x_k=\frac{1}{n!} $$ is given in a development like it's obvious, ...
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Given a general region, find the double integral bounded between $y = x$ and $y=3x-x^2 $

$$ J = \iint_R (x^2-xy)\,dx \,dy, $$ Suppose region R is bounded between $y = x$ and $y=3x-x^2 $ My attempt using vertical integration: $$ \int^{x=2}_{x=0} \int^{y=3x-x^2}_{y=x} \left({x^2-xy}\right)...
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How to construct double integral to find the area, given $\mathsf{D}= { \left( x,y \right): y \geq x^2-1 , y \leq 2x+2 , y \leq 3}$

I don't understand this topic honestly. I thought there is supposed to be two types of ranges for x and y. $$\therefore \space \mathsf{for} \space \space x^2-1 \leq 3 \space ; \space x \in [-2,2] \...
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Infinite sum of iterated integrals of matrix products

Edit: Discussion moved to Mathoverflow at https://mathoverflow.net/questions/395085/infinite-sum-of-iterated-integrals-of-matrix-products The problem: Let $$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \...
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Integral of $\int ^{5}_{1}\int ^{\sqrt{x-1}}_{y=0}{ye^{(x-1)^2}}dydx$

$\int ^{5}_{1}\int ^{\sqrt{x-1}}_{y=0}{ye^{(x-1)^2}}dydx$ First, I changed the integration from $x$ to $y$. However, I get $\int ^{5}_{1}\dfrac{\left( x-1\right) }{2}e^{\left( x-1\right) ^{2}}dx$ and ...
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Reverse the order of integration of $ \int_{-1}^1 \int_{x^2}^1 \sqrt{y}\ dydx$.

I am computing the following integral $$ \int_{-1}^1 \int_{x^2}^1 \sqrt{y}\ dydx$$ and I want to verify the exchanging of order of integration. The region is described as $-1\leq x \leq 1, x^2\leq y \...
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A question about iterated integral

In a circumstance I had to show that $$ \int_0^t \int_0^x \int_0^y f(z)\,dz\,dy\,dx=\frac{1}{2}\int_0^t f(z)(t-z)^2 \,dz $$ , where $t$ is a constant. I succeeded in doing so by integration by parts. ...
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$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$

I would like to solve the given integral: $$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$$ The integral is doable just using the regular iterated integral with $x$ and $y$. We integrate with respect ...
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Greens Theorem & does the perimeter of a double integral over a region converge to the perimeter of the original region?

When I've seen double integrals presented, usually its visualized as adding a bunch of small rectangular dA elements along the region. It feels pretty reasonable that this converges to the area. What ...
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Questions using Iterated Integrals.

so Ive just been introduced to the idea of iterated integrals and im finding it hard to work out how to complete questions on this subject and was wondering if anyone could help. So if I have an ...
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how to prove that two iterated integral are different

We have the following function: $$f(x,y)= \begin{cases} 1, & 0 \leq x-y \leq 1\\ -1, & 0 \leq y-x \leq 1 \\ 0, & \text{other cases} \end{cases} $$ and I need to prove that $...
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How do I get the normal domain from an inequality condition (application of Fubini's Theorem)?

I want to compute the volume of the 3-dimensional unit ball. $A := \{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \leq 1$ From this I want to get the normal domain as: $A = \{(x,y,z): x \in [-1,1], y \in ...
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Find the surface area of the sphere inside the cylinder

The given equations are of a sphere and cylinder respectively $$x^2+y^2+z^2=400$$ $$x^2+y^2=256$$ Solving the sphere equation for $z$ yields $$z=\sqrt{400-x^2-y^2}$$ Now to find $ds$ we take the ...
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2 votes
2 answers
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Problem with Infinite number of iterated integrals

I have been working on a problem from a past high school Math Competition and I have been stumped. I am asked to compute $$\lim_{n\rightarrow\infty} \int_{0}^{1}\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}\...
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Converting an iterated integral into a **sum** of iterated integrals

Question: Suppose $f$ is integrable. Express $\int^1_0 \int^{2y}_y \int^{x+y}_0 f(x,y,z)\ dz\ dx\ dy$ as a sum of iterated integrals in the order $\ dx \ dy \ dz$. My Attempt: Since $z=x+y$, $z_{max}=...
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continuity of a function and iterated integrals

Let $f:[0,1]\times [0,1] \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ defined as : \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} f(x,y) = \left\{ \...
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I have a question on Lebesgue Iterated Integrals

Let $(X, S, \mu)$ be a finite measure space. Suppose $g: X \rightarrow \mathbb{R}$ is integrable and that $g(x) \geq 0 \ \forall x \in X$. Let $A = \{(x,y) \in X \times [0,\infty) : 0 \leq y < g(x)...
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how to use iterated integral to find area of a region? [closed]

Use iterated integral to find area of a region bounded by $y=\sqrt{x}, y=3, x=0$, and $x=4$. sketch the region.
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Why are these two inequalities not the same even though they use the same equation?

I just don't get it, like at all. $U_{n}$ is an iteration defined on $\mathbb{N}$, BTW. The question was: $$\begin{align} U_{n+1} &= \frac{8U_n - 8}{U_{n} + 2} = 8 - \frac{24}{U_{n} + 2}\\ ...
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Am I putting the right limits of the integral?

Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} ...
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Proving the non-integrability of the following function on $(0,1)\times(0,1)$

So, basically, I've been trying for a while to prove that the function $f(x,y)=\frac{x-y}{(x+y)^3}$ is not integrable in $(0,1)\times(0,1)$. That is, I want to prove that the integral of it's absolute ...
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Problem with the existence of a iterated integral

Let $$ F(x,y)= \begin{cases} 1 &\mbox{ if } x\in\mathbb{Q},\\ 0 &\mbox{ if } x\in\mathbb{R}\setminus\mathbb{Q} \end{cases} $$ Then, $\int_{0}^{1}\int_{0}^{1}F(x,y)dydx$ exists?\ I'm convinced ...
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Showing Iterated Lebesgue Integrals Do Not Coincide

Let $f(x,y) = \frac{x}{1 - y^2}$ on $(-1,1) \times (-1, 1)$. I have two iterated Lebesgue integrals $$I =\int_{-1}^1 \int_{-1}^1 f(x,y)dxdy ~~~~~~~~~~~\text{and}~~~~~~~~~~~ J =\int_{-1}^1 \int_{-1}^1 ...
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I am unable to find correct answer after putting limits

$$ \int_0^3\int_0^{\sqrt{9-y^2}}y\,dx\,dy $$ I am able to evaluate the integral correct but when i put the limits on the last integral my calculation returns $0$, but the correct answer should be $9$....
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Riemann integration of a piecewise function over $E=[0,1]\times [0,1]$

For the following, let $E=[0,1]\times[0,1]$ and let $S$ be the subset of $E$ defined by $$S=\{(x,y)\in E \ | \ x=\dfrac{2j+1}{2^n}, y=\dfrac{2k+1}{2^n}, j,k=0,...,2^{n-1}-1, n\in \mathbb{N}\}$$ Let $...
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Why this iterated integral shouldn't start with $\Omega_1$ or end with $\Omega_0$ in order to converge

I have an issue with the following problem involving iterated integrals: Context. Let $(\omega_1, \dots, \omega_n)$ be complex-valued differential 1-forms defined in a real interval $[a,b]$. For all ...
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Given a vector equation, find the surface area

Question: r(u,v) = ucosv i + usinv j + v k for v and u, 0<=v<=7pi, 0<=u<=1 Find the surface area. from Google I got these steps: Find ...
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Evaluate the iterated integral

Evaluate the iterated integral: $$\int_0^2 \int_0^1 (2x+y)^8 \,dxdy$$ I tried doing a u-sub by letting u = 2x+y but then once I got to the outer integral I got $$\int_0^2 (2+y)^9/18 - y^9/18 \, dy$$ ...
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Converting Iterated Triple Integral from Cartesian to Cylindrical Coordinates

I am trying to convert the following iterated integral from Cartesian to Cylindrical coordinates: $$\int_{{\,0}}^{{\,\sqrt{3}}}{{\int_{{\,y}}^{{\sqrt {6 - {y^2}} }}{{\int_{{\sqrt \frac{{{x^2} + {y^2}}...
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Can we set up the volume of a 3D solid as a sum of two triple integrals?

For example, consider this question: Write a triple integral, including limits of integration, that gives the specified volume: Under the sphere $𝑥^2+𝑦^2+𝑧^2= 9$ and above the region between $𝑦=�...
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Is this the correct integral for finding the volume of the region bounded by the planes $𝑧= 3𝑦,𝑧=𝑦,𝑦= 1,𝑥= 1,$ and $ 𝑥= 2$?

The integral I've formed here is: $$\int_1^2 \int_0^1 \int^{3y}_y 1 dzdydx$$ Which evaluates to one. Not sure if I'm thinking of the region correctly, does this integral make sense for this region? ...
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Area between two polar curves using iterated integrals?

The question is from a practice exam I am currently trying to do: I am really not sure how to go about this one. In essence, I'd imagine that the idea is to find the area of the greater curve, and ...
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1 vote
1 answer
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Divergence of a double integral of a exponential

Problem: Solve the double integral $$I:=\int _{-\infty}^{\infty} \int _{-\infty}^{\infty} dx dy\ f(x,y),$$ where $f(x,y) = e^{-i a(x - y)}$ if $y > x$ and zero otherwise ($a$ is only a positive ...
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