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Yes, both methods approach the same limit. In this case, you can explicitly write that down: Assume you integrate from 0 to 1, and you split the range into N intervals of equal length. In the first case, your integral approximates as [math]I_{1, N} = \frac 1N \sum_{i=0}^{N-1} f(i/N) = \frac 1N \left( f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) \right)[/math]. In the second case, your integral approximates as [math]I_{2, N} = \frac 1N \sum_{i=0}^{N-1} \frac 12 \left( f(i/N) + f((i+1)/N) \right) = \frac 1N \left( \frac 12 f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) + \frac 12 f(1) \right) [/math]. If you compare the terms, you notice that [math]I_{1, N} - I_{2, N} = \frac 1N \frac 12 (f(0) - f(1) ) = \frac{f(0) - f(1)}{2N}.[/math] So whatever finite numbers f(0) and f(1) are, the difference between the two ways to approximate the integral becomes tiny when N becomes large enough. Btw: This editor is horrible: Preview should preview the rendered tex, not show me the raw tex I typed for different screen sizes. I want my editor from ten years ago back.

Even if one stipulates that politicians in the US are religious, clearly the other answer is the US is a representative governance system and voters feel those representatives represent them. That said, I tend more to agree with swansonts core point here. One could say he represents my view. There’s a certain beauty in this realization. It means we’re lucky and have a chance to explore and learn and be a part of something bigger than ourselves, even if only temporarily. If that depresses you, you should consider seeking help from a medical professional. ‘We can lament the fact that rose bushes have thorns, or we can rejoice about the fact that the thorn bushes have roses.’ Perspective.

They aren’t. They pretend to be, on one or two issues, in order to pander to their base.

And for zinc- which is often used in school experiments on photoionisation, the reflectance drops like a rock in the UV region https://tubingchina.com/HDG-Hot-Dip-Galvanized-Surface-Reflectivity.htm

Is not a common recipe

So there is a view that metabolites of the acetyl coA pathway are ancient. However, there is also the discovery that certain alloys can convert hydrogen and carbon dioxide into formate, acetate and pyruvate. I.e. they can be formed without enzymatic activities which would in theory allow organisms to utilize them, without having any producers first. (Preiner et al. 2020; https://doi.org/10.3390/life8040041)

Exactly as @exchemist says. It's not that if you shoot very energetic photons against hydrogen atoms the photoelectric effect doesn't go on. It does. But the clever trick is to use a metal, because there you can show that no matter how many photons you shoot against the metal, they won't free electrons from the metal unless they have the required frequency (energy = h x frequency). They would just be absorbed by the continuum spectrum (available energies) of the metal. And that threshold energy is nicely shown in a metal because there is a sharp gap of energy that the electrons have to surmount if the are to be kicked off from the metal. So the metal: 1) Completely absorbs any photons below the threshold kick-off energy 2) Emits electrons when the frequency surpasses that threshold energy They act like a very efficient switch for the photoelectric effect.

The result of the integral doesn't depend on which approximation you use. The second one is called the lower Riemann sum. There is another one with starts with what you would call \(f\left(x_{0}+\triangle x\right)\) instead of \(f\left(x_{0}\right)\), and ends with \(f\left(x_{1}\right)\) instead of \(f\left(x_{1}-\triangle x\right)\) It's called the upper Riemann sum. Your expression differs only in a second-order term in \(\triangle x\). You only see a big difference because your \(\triangle x\) is enormous in the image. You can actually do an even better fit by taking a polygonal approach to the curve (for the same step \(\triangle x\).) https://www.geogebra.org/t/upper-and-lower-sum?lang=en Sorry. This is the applet that I meant to show you. You must play with the n=10. Take it up to n=24, for example, and you'll see what I mean. https://www.geogebra.org/m/SNS8SYSg

To create plasma arc use Van de Graaff generator. This is from my own: Alternatively you can build Cockcroft Walton generator https://en.m.wikipedia.org/wiki/Cockcroft–Walton_generator

Nice explanation. You can also use semiconductors etc to create a voltage with the photoelectric effect. This is called photvoltaics. https://en.wikipedia.org/wiki/Photovoltaics Yes this is how technologists are trying to build a fusion reactior. Essentially a plasma arc is initiated and contained in a magnetic 'bottle' long enough for the fusion process to start and become self sustaining. https://en.wikipedia.org/wiki/Tokamak

There is a useful article here about the care needed when interpreting raw numbers from VAERS: https://www.ibtimes.com/over-900-died-after-receiving-covid-19-vaccine-experts-say-data-misinterpreted-3153820 So the answer I think is no, they have not been investigated for possible causality, nor are there necessarily grounds for doing so. By the look of it, it may be the old post hoc ergo propter hoc trap. It seems quite hard to find the data you are looking for viz. the risk of fatal side effects of the vaccine in fit and healthy under 30yr olds. I have been able to find the relative risk due to the virus of death, hospitalisation by age bands in the US, but this does not filter out the fit and healthy from the others: https://www.cdc.gov/coronavirus/2019-ncov/covid-data/investigations-discovery/hospitalization-death-by-age.html I suspect that, for the cohort you are interested in, the main issues will be (i) the risk to them of Long Covid from the virus, and (ii) the increased spreading of the virus in the population by this group if they are not vaccinated - and hence the potential for further variants to arise. I also suspect we simply do not have data on all this, given that the disease has only been with us for 18 months and given the huge number of variables, e.g. there are more than 5 vaccines in widespread use, there are half a dozen variants of the virus, all with their own characteristics, etc. The only further thing I can contribute is that, in the UK, people under 30 are recommended not to have the Astra/Zeneca vaccine but one of the others, due to the low but finite risk of blood clots. I would think the same applies to the Johnson & Johnson one, as that, I gather, uses similar technology. So Pfizer and Moderna seem to be fine for younger people. And by being vaccinated you do your bit for society by stopping the spread and reducing the chance of further lockdowns due to new variants. Plus you avoid the risk of long Covid, like my 25yr old nephew, and which I myself have had a brush with, as a fittish 65yr old.

@Alfred001If I may express what is implicit in the foregoing responses in a much more vigorous form: go ahead. Don't take the vaccine. Let people know you have not taken the vaccine. Wear a badge clearly stating you have not taken the vaccine. This will allow the rest of us to identify you as ignorant and anti-social. We can then respond accordingly. Alternatively, take the vaccine, thereby improving your chances of avoiding problems, showing your grasp of reality and making an important contribution to community welfare.

You really have no clue what you're saying do you.