Trapezoidal rule that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as "Monkhorst-Pack integration". For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule. The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has
Zaitsev's rule the University of Kazan in 1873. Zaitsev had cited Popov's 1872 paper in previous work and worked at the University of Kazan, and was thus probably aware of Popov's proposed rule. In spite of this, Zaitsev's 1875 "Liebigs Annalen" paper makes no mention of Popov's work. Any discussion of Zaitsev's rule would be incomplete without mentioning Vladimir Vasilyevich Markovnikov. Zaitsev and Markovnikov both studied under Alexander Butlerov, taught at the University of Kazan during the same period, and were bitter rivals. Markovnikov, who published in 1870 what is now known as Markovnikov's rule, and Zaitsev held conflicting views regarding elimination