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 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.

Simpson's paradox for continuous data: a positive trend appears for two separate groups ( blue and red ), a negative trend ( black, dashed ) appears when the data are combined.

( 1999 ) also criticised Van Howe's paper, stating that his results were a case of " Simpson's paradox, which is a type of confounding that can occur in epidemiological analyses when data from different strata with widely divergent exposure levels are combined, resulting in a combined measure of association that is not consistent with the results for each of the individual strata.

When using Simpson's Diversity Index on 10 aggregated ethnic groups, the 2001 census identified Newham as the most ethnically diverse district in England and Wales, with 9 wards in the top 15 .< ref >< cite > Simpson's diversity indices by ward 1991 and 2001-GLA Data Management and Analysis Group ( page 11, Table 3 ) Greater London Authority, January 2006 ), accessed 13 December 2006 </ ref > However, when using the 16 ethnic categories in the Census so that White Irish and White Other ethnic minorities are also included in the analysis, Newham becomes the 2nd most ethnically diverse borough < ref >< cite > Focus on Newham 2006-local people and local conditions-London Borough of Newham, Corporate Research Unit ( chapter 2, page 24 ) ( 2006 ) date accessed 31 March 2007 </ ref > with 6 out of the top 15 wards, behind Brent with 7 out of the top 15 wards.

In 1871, Richard Simpson proposed that some additions to the play had been written by Shakespeare, and a year later James Spedding, editor of the works of Sir Francis Bacon, while rejecting some of Simpson's suggestions, supported the attribution to Shakespeare of the passage credited to Hand D. In 1916, the paleographer Sir Edward Maunde Thompson published a minute analysis of the handwriting of the addition and judged it to be Shakespeare's.

The New York Times acclaimed Simpson's analysis of how sport and advertising are both increasingly using homoerotic imagery, in a process he dubbed " sporno " (" the place where sport and porn meet and produce a gigantic money shot ") as one of the Ideas of the Year.

The colts in Simpson's stable have little if anything on the fillies, especially the pacers.

Maris purchases $100 suits from Simpson's in New York.

The allegations of seatbelt failure resulted in Bill Simpson's resignation from the company bearing his name, which manufactured the seatbelts used in Earnhardt's car and nearly every other NASCAR driver's car.

Simpson's department stores, which were acquired by Hudson's Bay Company in 1978, were converted to The Bay stores in 1991.

Kemp supported rights for illegal immigrants, but opposed Lamar Smith and Alan Simpson's proposed restrictions on legal immigration.

Ribbentrop's time in London was also marked by rumors that he was having an affair with Wallis Simpson, British-businessman Ernest Simpson's wife and King Edward VIII's mistress.

The Americans believed that Ribbentrop used Simpson's access to the King to funnel important information about the British to his superiors.

Simpson's murdered ex-wife, Nicole Brown Simpson.

Paradoxes in economics tend to be the veridical type, typically counterintuitive outcomes of economic theory, such as Simpson's paradox.

" Simpson's conjecture was that according to the geological record, on very rare occasions evolution would proceed very rapidly to form entirely new families, orders, and classes of organisms.

Simpson's idea was principally concerned with evolution at higher taxonomic groups.

Plate from Christopher Simpson's book, The Division Violist, England, 1659 – 1667 edition.

While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

Peter Simpson's translation, with Greek terms inserted in square brackets.