If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

Numerical calculation of the integral using the trapezoidal rule or Simpson's rule is not efficient and becomes numerically unstable especially for configurations with large Fresnel number.

Illustration of Simpson's rule.

Simpson's rule, which is based on a polynomial of order 2, is also a Newton – Cotes formula.

For instance, the standard fourth-order Runge – Kutta method applied to the differential equation yields Simpson's rule from above.

The weights are chosen such that if is independent of, so that the differential equation is equivalent to a simple integral, then RK4 is Simpson's rule.

Simpson's rule can be derived by approximating the integrand f ( x ) ( in blue ) by the quadratic interpolant P ( x ) ( in red ).

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals.

Simpson's rule also corresponds to the 3-point Newton-Cotes quadrature rule.

Simpson's rule is a staple of scientific data analysis and engineering.

Simpson's rule can be derived in various ways.

Another derivation constructs Simpson's rule from two simpler approximations: the midpoint rule

This weighted average is exactly Simpson's rule.

This yields Simpson's rule.

The error in approximating an integral by Simpson's rule is

Simpson's rule gains an extra order because the points at which the integrand is evaluated are distributed symmetrically in the interval b.

Since the error term is proportional to the fourth derivative of f at, this shows that Simpson's rule provides exact results for any polynomial f of degree three or less, since the fourth derivative of such a polynomial is zero at all points.

If the interval of integration is in some sense " small ", then Simpson's rule will provide an adequate approximation to the exact integral.

For such a function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

In these cases, Simpson's rule may give very poor results.

Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval.

* O. J. Simpson murder case – O. J. Simpson's trial, described in the U. S. media as the " trial of the century " and enormous U. S. media attention is focused on the trial.

Quantum evolution is a component of George Gaylord Simpson's multi-tempoed theory of evolutionary change, proposed to explain the rapid emergence of higher taxonomic groups.

In probability and statistics, Simpson's paradox ( or the Yule – Simpson effect ) is a paradox in which a correlation present in different groups is reversed when the groups are combined.

Though it is mostly unknown to laypeople, Simpson's Paradox is well known to statisticians, and it is described in a few introductory statistics books.

Since Edward Simpson did not actually discover this statistical paradox, some writers, instead, have used the impersonal names reversal paradox and amalgamation paradox in referring to what is now called Simpson's Paradox and the Yule-Simpson effect.

She is the teacher of Bart Simpson's 4th grade class at Springfield Elementary School.

It concludes that Riviera is a better role model for physicians, whereas Hibbert is a paternalistic and wasteful physician, unlike Riviera, who strives to cut costs and does his best to avoid the coroner, disproving Marge Simpson's claim that he is not a doctor at all.

He is best known for being the most frequent recipient of Bart Simpson's prank calls.

He is the town drunk and Homer Simpson's friend.

After an attempt is made on Homer Simpson's life, Bob is released from prison to help find the culprit.

The Upminster ward of the borough is the least ethnically diverse in Greater London, with a Simpson's diversity index of 1. 05.

Huxley's book is not strong in palaeontology, which illustrates perfectly why Simpson's later works were such an important contribution.

A Simpson's Comics Treehouse of Horror issue features a parody of this episode, which is also based on the classic movie Invasion of the Body Snatchers.

Fuhrman stated he found a blood-stained glove at Aaron Turner's condo ( the scene of the murders of Nicole Brown Simpson and Ron Goldman ), and also found another at Simpson's home.

This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding:

Simpson's 3 / 8 rule is another method for numerical integration proposed by Thomas Simpson.

Simpson's childhood, growing up, getting climbing experiences and also the period after those described in Touching the Void ( including the part about the complicated healing of his injuries and writing the book ) are described in another of his books, called This Game of Ghosts ( Vintage Books, 1993 ).

The inter-war years saw another high period for Simpson's.

In 1959, Simpson's lost another notable member of its staff.

As with many other works by Nielsen starting as early as his first symphony, this symphony uses " progressive tonality ", not only starting in one key — G, here — and ending in another ( B-flat ) but making the change part of the drama of the work ( this was one of the main theses of Simpson's book ).