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If Integration is about finding the Area under the curve, how can the Simpson Rule use curved Parabolas to find Area under the slice?

I understand the Trapezoid rule because it uses triangles which are made of straight lines, but how can the Simpson rule use Parabolas if they themselves are curved?

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    $\begingroup$ What is the problem with using parabolas? You can use anything you want whose area you know. It's not as if we must avoid knowledge of how to integrate simpler functions (like parabolas) to get methods of numerically estimating integrals of more complicated functions. There is no law that one must avoid using the Fundamental Theorem of Calculus here. $\endgroup$
    – KCd
    Dec 2, 2017 at 4:04
  • $\begingroup$ singerng has given a good answer to the question you asked. The next question is why would we use parabolas instead of trapezoids? Parabolas make the answer exact for functions that are up to cubics (quadratics because parabolas match that and cubic because of symmetry) while trapezoids are only exact for linear functions. For functions that are not polynomials, Simpson will be more accurate if the function is well approximated by a cubic. If not, the shorter intervals of the trapezoid rule are important. "HIgh order is not always high accuracy" $\endgroup$ Dec 2, 2017 at 5:12
  • $\begingroup$ @KCd True I hadn't thought of that. I was operating under the assumption that we can only integrate by simplifying the curve into slices of things that we already know the exact area of. But as you said, using FTC, we could still get those exact slices. $\endgroup$
    – ruckarucka
    Dec 2, 2017 at 8:09
  • $\begingroup$ @ruckarucka it was known before there was a Fundamental Theorem of Calculus. See en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula. $\endgroup$
    – KCd
    Dec 2, 2017 at 11:34

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The trapezoid rule works by approximating any given function, over small enough intervals, as lines, and then adding up the areas of the resulting trapezoids. The idea behind Simpson's rule is the exact same: just as we can approximate over a small interval as a line, so can we approximate as a parabola. As long as we know the formula for the area under a section of any parabola (which has been well-known since antiquity), we can approximate the total area under the curve by slicing it up and adding up areas of small parabolas. Lines are curves too, but they need two points to "fix" them; quadratics need three points, but otherwise they're just like lines. Just like adding up areas of trapezoids, we can also add up areas of parabolas, and it works even more accurately.

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  • $\begingroup$ Simpson's rule is more accurate because weird, curvy functions look more like parabolas than lines. $\endgroup$ Dec 2, 2017 at 5:12
  • $\begingroup$ Huh, I didn't know that the area under a parabola was already known. That was the source of my confusion because I thought you would have to use integration to find area under a parabola which would make Simpson's rule a self-defeating solution. Thank you for clearing that up. $\endgroup$
    – ruckarucka
    Dec 2, 2017 at 8:01
  • $\begingroup$ Well, we don’t use these techniques necessarily to prove what the values of various integrals are (we can simply take antiderivatives if we have the Fundamental Theorem of Calculus). We use them to approximate the values of integrals for complicated functions. $\endgroup$ Dec 2, 2017 at 13:07

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