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For the Riemann integral, are there any methods of numerical integration that do not involve rectangles or approximating the area with a polynomial function? I am aware of the trapezoidal rule, but I am considering it to fall under rectangles, as it is the average the sum of the left and right Riemann sums. More specifically, are there any numerical integration methods based on partitioning the area under the curve into shapes other than rectangles (especially ones that involve other geometric shapes)?

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  • $\begingroup$ Gaussian Quadrature approximates the integral by doing piecewise interpolation with a polynomial en.wikipedia.org/wiki/… $\endgroup$
    – irchans
    Jan 15, 2019 at 19:20
  • $\begingroup$ See whether "Monte Carlo integration" meets your criterion. $\endgroup$ Jan 15, 2019 at 19:21

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Simpson's rule can be understood as partitioning the area under the curve into the areas under segments of parabolas.

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    $\begingroup$ It appears to me that the heart of Simpson's rule is more about polynomial approximation than actually partitioning the area. $\endgroup$
    – H Huang
    Jan 17, 2019 at 1:29

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