Quadrature refers to techniques in numerical integration, such as Riemann sum approximations, Simpson's rule and Gaussian quadrature.

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### Ancient Greek proofs of Archimedes' three properties of the parabola?

Please refer to the document, "Archimedes' Quadrature of the Parabola": https://www2.bc.edu/mark-reeder/1103quadparab.pdf This document describes how Archimedes proves that the area of any parabolic ...
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### Why not simply use sine weights with Clenshaw-Curtis nodes?

Clenshaw-Curtis quadrature is based on writing $$\int_{-1}^{1} f(x)dx=\int_{0}^{\pi}f(\cos y)\sin y dy$$ and then replacing $f(\cos y)$ by a truncated Fourier series, so that the integral can be ...
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### the highest degree of the polynomial, for which the above formula is exact?

Consider the quadrature formula $\int _{-1}^1\vert x \vert f(x)dx \approx \frac{1}{2}[f(x_0)+f(x_1)]$, where $x_0$ and $x_1$ are quadrature points. Then the highest degree of the polynomial, for which ...
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### Cubic Bézier curve arc length parametrization reversal: find t given a length

I am following this paper Approximate Arc Length Parametrization, M. Walter & A. Fournier, 1996 and have succesfully implemented the direct solution, as in finding the length $s(t)$ given $t$. ...
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### References on quadrature of $\int_0^{\infty} f(x) \exp(-x^a) dx$

I am aware of Hermite-Gaussian quadrature techniques for integrals of the form $$\int_0^{\infty} f(x) \exp(-x^2) dx$$ However, am I looking for references on quadrature where the exponent is more ...
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### Deriva parameters of Gauss quadrature rule

Derive the parameters of Guass quadrature with three points $$\int_{-1}^1 f(x)\,\mathrm dx \approx C_1 f(\xi_1)+C_2 f(\xi_2) + C_3 f(\xi_3)$$ such that the integral is exact up to $x^5$. ...
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### High accuracy root finder of Legendre polynomials' derivatives?

I need GLL (Gauss-Legendre-Lobatto) nodes for the Legendre-Galerkin-NI spectral method. It requires me to find the roots of the derivatives of Legendre polynomials. My Matlab program calculates the ...
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### Approximation to $\int_{-1}^{1}f(x)dx$ using only $f(0)$, $f'(-1)$, and $f''(1)$

The exercise is Using only $f(0)$, $f'(-1)$, and $f''(1)$, compute an approximation to $\int_{-1}^{1}f(x)dx$ that is exact for all quadratic polynomials I have only seen guassian quadrature ...
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