How does numerical integration work in programming?

Since numbers in the computer are discrete, I was wondering how can a computer calculate the numerical integration of a function? Thanks.

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An article I wrote for the r/roblox tech-blog about numerical methods and integration of the motion of spinning bodies. robloxtechblog.com/motion…
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Does numerical integration require a continuous function?

Simple question, if you have a set of data points plotted in 2D, can you numerically integrate them? Or do you need to fit a function to these data points before you can numerically integrate?

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Wrote this article on spinning rigid bodies and numerical integration (for Roblox Tech Blog). Delves into mathematical details of simulating rotational motion of rigid bodies, mainly in computer game physics engines. robloxtechblog.com/motion…
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Can't figure out B to this Simpson's Rule Numerical Integration Problem. I have part A completed, but I don't understand how the series given in part B relates to part A. A hint was given that we should use the "composite trapezoid rule". I just don't see how we can find N for a number as big as 911 reddit.com/gallery/jzbod2
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Getting Initial Conditions from Numerical Integration Results

Let’s say you have some initial conditions for a set of ODEs and you need to do some numerical integration. Once you have those results, is there a method to back out what your initial conditions were?

I’ve been out of school for a while so I am sorry if my questions is basic or I haven’t worded it correctly.

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The common Numerical Integration techniques.
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|Numerical Integration| What is Simpson's Rule? [Intuition] :) v.redd.it/m572spvr18r41
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[F#] Numerical Integration in 30 lines, plus 100 lines of tests

I am a mathematician by profession, and am trying my hand at F#.

This is not my first exercise, but I would like some feedback especially on the testing part - I'll soon be building a much bigger system for more sophisticated numerics.

Pastebin: https://pastebin.com/R6z08kER

Gist: https://gist.github.com/OrlandoRiccardo/543fb187e6f00589a7f99cbd8884623b

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[F#] Numerical Integration in 30 lines, plus 100 lines of tests /r/CritiqueMyCode/comment…
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[College math: Numerical integration] How can I remove singularities from integral?

I've got an integral from 2 to infinity of 1/[x*ln^2(x+sqrt(x))] to integrate numerically using matlab. First I removed singularities and made it into a proper integral by substitution y=1/x, which gave me integral from 0 to 1/2 of 1/[y*ln^2(1/y+1/sqrt(y))], which still have a singularity and I don't know how to remove it in order to integrate this numerically. Can you help me with this?

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AP Calc: Numerical Integration with Tables

Self studying BC here. For some rather inexplicable reason, I just suck at numerical integration (Riemann sums, midpoint, trapezoidal, etc). This is specifically the case when the info is in tables with unequal intervals. It always costs me two points when it comes up. I've looked for resources but there's nothing specifically geared to applying those techniques with tabular data. The FRQ solution guides don't make much sense to me.

Could someone give me some resources to work with or guide me through it with an example? That would be much appreciated.

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A Video Lecture on Numerical Double Integration
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Numerical Integration of Differential Equations

When can a differential equation be solved through analytical integration, and when can it be solved ONLY through numerical integration. Or in other words: What boxes must a differential equation need to "tick" before I know it can be analytically integrated?

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Numerical integration is better in certain situations.
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Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration?
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Error bounds of numerical integration

I am trying to understand how to use the error bound of a trapezoidal numerical integration.

If the error bound is

abs(E) <= (K(b-a)^3 ) / (12n^2)

and

f''(x) <= K

How do I find (or reasonably estimate) K if I don't have the exact f(x) ?

Thanks.

Edit: To elaborate, I'm trying to propagate errors through a rather complex equation that includes a trapezoidal integration. I have the error due to my 6 measured variables accounted for, but I haven't been able to figure out how to account for the error to to the numerical integration. Any help on this would be greatly appreciated.

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Numerical Methods in C++ Part 1: Newton-Cotes Integration thoughts-on-cpp.com/2019/…
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Very weird results with numerical integration using trapezoid method - Online Technical Discussion Groupsβ€”Wolfram Community

The graph I got using the code in the link is VERY peculiar to say the least. I have no idea what's causing it.

https://community.wolfram.com/groups/-/m/t/1840568

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Numerical integration and optimization

I want to solve something like min_x \int f(x,y)dy automatically. Not sure if there's a way to do this in JuMP with a numerical integration method or if I have to write out the integral approximation.

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Been learning numerical integration recently
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Numerical Methods in C++ Part 2: Gauss-Legendre Integration thoughts-on-cpp.com/2019/…
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[1902.00648] Numerical Integration Method for Training Neural Network arxiv.org/abs/1902.00648
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Is there a way to express the position of a celestial body in orbit in r(t) and ΞΈ(t) without numerical integration?

As far as I know, there's no mathematical way to analytically express the position of a celestial body orbiting another body in a two-body problem at time t, just like calculating precisely the circumference of an ellipse.

Am I wrong? If I'm right is there an equation that approximates r(t) and ΞΈ(t)?

If it helps, I'm trying to plot the density wave theory for my students so they can see where the individual stars go.

Edit: I'm talking about elliptical orbits because there are no perfectly circular and parabolic orbits in real life.

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