# Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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### Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
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### Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
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### Integrability property of polynomials in several variables

This might be very trivial, or not. Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
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### Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
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### A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
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### Hilbert series for invariant ring

I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with). To put it in some broader context, let $G$ ...
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### An integral for the tribonacci constant and the general case

When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer, $$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}}$$ However, the ...
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### Area of generalized ellipse

An ellipse $E$ can be defined by two foci, $p,q\in\mathbb{R}^2$, and a length parameter $\ell$ as follows: $$E = \{x\in\mathbb{R}^2 : ||p-x||+||q-x||\le\ell \}.$$ The area of $E$ is uniquely ...
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### Integral-like concepts

I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect. The background is that I have proven a ...
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### Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
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### Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
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### Is the following integral positive or not?

Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...