# 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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### Two double integrals (in terms of polar) producing the same result?

I was working on a problem where I had to convert the region into polar equations in order to find the double integral. The region was $$r = 2\cos(\theta)$$ and after solving the problem I realized ...
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### Solve $\int_{0}^{1}\int_{0}^{1-y} xy\sqrt{(1-x-y)}dxdy$ [closed]

Calculate the double integral $$∫∫xy\sqrt{1-x-y}\,dxdy$$ where the domain is $D=\{(x,y):x≥0,y≥0,x+y≤1\}$ I think the range is $0≤x≤1$ and $0≤y≤1−x$ . Is it correct? I understands that this problem ...
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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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### 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? ... 151 views

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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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### 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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### 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 ... 