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

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### Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$). So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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### When can one trust that an adaptive quadrature scheme is not missing nontrivial areas of the domain?

hcubature http://ab-initio.mit.edu/wiki/index.php/Cubature_(Multi-dimensional_integration) seems to be a useful package for computing multidimensional integrals numerically. However, I have come upon ...
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### Why are isolated zeros allowed in the weight function?

In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function. Then there is the following question: Explain why isolated zeros are allowed ...
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### Gaussian Quadrature and Degree of Precision

how to prove a interpolatory quadrature has degree of precision k = 2n + 1 if and only if it is a Gaussian Quadrature
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### Roughness of $\int\sqrt{1-t^2}f(t)\,dt$ in $[-1,1]$

Why is this integrand non-smooth in $[-1,1]$? $$\int\sqrt{1-t^2}f(t)\,dt$$ where $f$ possesses an analytic extinction to a complex neighbourhood $[-1,1]$ More general, how can I find it out? Thanks ...
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### Determining constants to make a quadrature formula with degree of precision equal to three.

I just need an explanation on how these linear equations are found when plugging and testing $f(x) = x^k.$ For example, in the first linear equation, what happens to the constants c and d. Why are ...
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The problem states the following: Find with at least 10 digitis of precisión the roots of the following equation: $\int_x^{x^2} \!e^{-t^2}\,\mathrm{d}t = x^5 -3x^2 + 1$ in the closed Interval [-1,1]. ...