Are there any numerical integration methods that do not involve rectangles or polynomial approximations?

Question

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)?

Answer 1

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