Comment (by Zach Etienne):

A correctly implemented extended midpoint rule should be 2nd-order accurate (Numerical Recipes, Section 4.1), with errors dropping like 1/N^2, where N is the number of subintervals. In addition, on spheres midpoint rule is often preferred as it can avoid sampling on the singular z-axis.

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