# Questions tagged [iterated-integrals]

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70 questions
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### Application of the Divergence Theorem with change of variable

Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
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### Show Lebesgue Integrable and Compute the Two Iterated Integrals

(I am working on problems having to do with Fubini's Theorem) Given $α ∈ (0,∞)$, show that the function $(x, y) \mapsto e^{−αxy}\cdot sin x$ is Lebesgue integrable on $(0,∞) × (1,∞)$. Compute the two ...
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### Wedge of Cheese Iterated Integral

I am stuck on a practice exam problem. A wedge of Manchego can be modeled as the solid region in $R^3$ bounded by the following surfaces: z = 0, z = y, y = 9-$x^2$. Express the volume of this ...
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### Estimate for multiple harmonic sum

I am interested in estimating the following family of sums: $$S_k(n) \equiv \sum_{\substack{n_1, \ldots, n_k \geq 1\\n_1 + \ldots + n_k = n}}\frac{1}{n_1\ldots n_k}$$ where $k \geq 1, n \geq 1$. A ...
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### Differentials and Integration

I have been informed that consecutive differentials in iterated integral problems are actually connected via the exterior product. So the factor $dx\ dy$ in $\int\int x^2\ dx\ dy$ is actually the ...
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### Find $\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$

$$I:=\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$$ Here the region of integration is the triangle with vertices $(0,0),(0,1)$ and $(1,1)$ and given as a type-1 region. We can convert it into a type-2 ...
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### Integral $\int_{[1,2]\times[0,\pi]}\log(\sqrt{x})\sin(2y)d(x,y)$

I want to find out if this integral can be calculated (if it exists) $$\int_{[1,2]\times[0,\pi]} \log(\sqrt{x})\sin(2y)~d(x,y)$$ To be honest, I don't know how, but I think that one might has to use ...
1answer
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### Integral $\int_{[0,1]^3}^{} (x^3+y^2)z^{-1}d(x,y,z)$

I want to find out if this integral can be calculated (if it exists) $$\int_{[0,1]^3}^{} (x^3+y^2)z^{-1}d(x,y,z)$$ To be honest, I don't know how, but I think that one might has to use Fubini's ...
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### Help in numerical verification of Chen's identity in rough path theory.

Would anyone please help me to verify Chen's identity as claimed in https://en.wikipedia.org/wiki/Rough_path#Signature for a simple two dimensional path? Consider a path between $a$ to $d$ and the ...
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### How do I perform a change in order of integration here?

I have a function $$\int^1_{y=0}\int^1_{x=y}e^{x^2}dx\ dy$$ Which I want to perform a change in order of integration. I have plotted the graph: And it seems it's the area bounded by the y-axis and x-...
1answer
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### Solve: $\iint_{B}\frac{\sqrt{y-x}}{1+y+x}\,dx\,dy$

B corresponds to the triangle with vertices (0,0), (1,0) and (0,1) I believe I am supposed to use variable substitution; however, whenever I can find a way to simplify the function, the integration ...
1answer
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### How can I solve this Iterated integral

domain E I've this integral on the domain E: x[0,1], y=x^2 $\int_E x^3sin(xy)\, dxdy$ I dont know how can I solve this through the two different ways:first integration in dx and then in dy, and ...
1answer
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### Evaluating $\zeta(4)$ using iterated integrals

I'd like to evaluate $\zeta(4)$ using iterated integrals. We already know the numerical answer, so it remains to set up the integral and do some of the steps. From the recipe of Ihara-Kaneko-Zagier ...
1answer
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### Why is this computation of the surface area incorrect?

So I am asked to find the surface area of the part of the curve $$x=5y+z^2$$ that lies between the planes $y=0,y=z, z=0,$ and $z=2$. I parametized my curve as $r(y,z)=\left<5y+z^2,y,z\right>$, ...
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### Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$

Reference:(Fubini's Theorem) Question: Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$ ? Edit: Both ...
1answer
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### Iterated logarithm rule general version exists?

Given a standard random walk, the Iterated logarithm rule say that with probability one, $$\frac{|w(n)|}{\sqrt{n \log\log n}}$$ has $\limsup$ $\sqrt{2}$ as $n \to\infty$. What about other values? ...
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### Problems with double integration

I'm trying to integrate $\int_{-a}^a\int_b^cy^{2m+1}e^{xy^{2n}}dxdy$.But I have never seen an integral with so many parts to it and I am little overwhelmed. How do I solve this?
2answers
81 views