# Questions tagged [approximate-integration]

Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

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### Midpoint rule integration for a matrix-vector product

Consider a function $F=F(q)$ which is a symmetric, positive-definite matrix form of dimensions $n\times n$, where $q \in \mathbb{R}^n$ and $q=q(t)$. When applying the midpoint integration rule on $F$, ...
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### Is the Monte Carlo integration method(s) actually a viable way to accurately integrate functions?

Recently, I was playing around with some Monte Carlo simulations using Python to evaluate integrals of functions such as f(x) = x(1-x)sin²[200x(1-x)]. I am aware that Monte Carlo integration methods ...
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### How to integrate arbitrary discrete-time linear and angular body fixed velocity to world space?

I have body fixed angular velocity values and linear acceleration values streaming in to my application. at some interval $\delta t$. I need to get a world position from these, assuming the start ...
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### Iterative formula for the integration of autonomous differential equations

Consider the following autonomous differential equation: $$y'=f(y)$$ where $y=y(t)$ with $y_0 = y(0)$. Let us suppose that $y^{[0]}(t)$ is a good approximation for $y(t)$. Using this function in the ...
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### Do numerical integration methods ever utilize evaluations of the derivative of the function?

Suppose one is performing numerical integration of some function $f(x)$, but in addition to being able to evaluate its value at points $f(x_1), \dots, f(x_n)$, one also can additionally obtain its ...
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### Proof that the Trapezoidal Method for solving IVP is $O(h^3)$

My question comes from Chapter 6 of the book Introduction to Numerical Analysis, 2nd edition, by Kendall Atkinson. In section 6.5, p.368-369, the author proves that the Trapezoidal method for solving ...
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### numerical spherical integration

I am trying to find a numerical sperical integration methods with high precision. I have been using the numerical integration based on lebedev quadrature of 131th order. Is there any numerical code ...
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### Determine whether rectangle/trapezoid area will be "overall" underestimate or overestimate in an interval

When estimating the area under a curve using left-bound rectangles, I know there will be an underestimate in a given interval if the function is increasing, $f'\left(x\right)>0$, and an ...
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### How can I (analytically or numerically) integrate the product of exp(-x) and fractionally degreed polynomials?

Are there any closed form solution to this integral? If not are there any good numerical methods to evaluate it? $$\int_{0}^{\infty} (x+3)^{2.5} \times (1+x)^{3.5} \times exp(-x) dx$$ I have tried ...
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### Approximation of integration of piecewise continuous function at discrete points

I have an integral of the form $$\int_{0}^{1} f(x)dx,$$ where $f$ is a continuous function that is not differentiable at countable number of points, defined on $[0,1]$. ...
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### Calculate the integrals $\int\limits_{0}^{\infty }e^{-\lambda (x-1)^2}dx+\int\limits_{0}^{\infty }e^{-\lambda (x-2)^2}dx$

In the $\lambda \to \infty$ limit, approximate the integral $$I(\lambda )=\int\limits_{0}^{\infty }e^{-\lambda (x-1)^2(x-2)^2}dx$$ I understand that the function $-\lambda (x-1)^2(x-2)^2$ reaches a ...
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