# How can Simpson Rule use curves to find the Integral?

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?

• 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.
– KCd
Dec 2, 2017 at 4:04
• 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" Dec 2, 2017 at 5:12
• @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. Dec 2, 2017 at 8:09
• @ruckarucka it was known before there was a Fundamental Theorem of Calculus. See en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula.
– KCd
Dec 2, 2017 at 11:34