Comment (by Zach Etienne):

The midpoint rule yields errors that scale with the square of the grid spacing; that is, for sufficiently smooth functions, the error should decrease proportionally to 1/n^2, where n is the number of intervals. See, e.g., https://en.wikipedia.org/wiki/Riemann_sum .

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There are higher order midpoint rules, but they require n to be divisible by e.g., 4.

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Ticket URL: https://bitbucket.org/einsteintoolkit/tickets/issues/2282/gallery-examples-use-low-order-integration