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.

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.

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.

This works to the players advantage, as it can be used to send up barrels of nuclear waste, which destroys the space ship, taking Homer Simpson's ' 70's Sports Car ', Snake and Bandit, John Frink and the Hover Car

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.

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

In 1961, Brian's largest surviving work, the Gothic Symphony, which had been written between 1919 and 1927, was first performed at Westminster Central Hall, in a partly amateur performance conducted by Bryan Fairfax, and in 1966 the first fully professional performance was given at the Royal Albert Hall conducted by Boult, both occasions largely the result of Simpson's lobbying.

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, of which the Low birth weight paradox is an example

* Application of Simpson's Rule-Earthwork Excavation ( Note: The formula described in this page is correct but there are errors in the calculation which should give a result of 569m3 and not 623m3 as stated )

On television, Fierstein was featured as the voice of Karl, Homer Simpson's assistant, in the " Simpson and Delilah " episode of The Simpsons and the voice of Elmer in the 1999 HBO special based on his children's book The Sissy Duckling, which won the Humanitas Prize for Children's Animation.

Nick Lachey was featured on Jessica Simpson's album on the track " Where You Are ", which was released as a single and reached the top 40.

The Rock was the last film in which Bruckheimer collaborated with Simpson, due to Simpson's untimely death.

Simpson College enjoys a historic close partnership with the Des Moines Metro Opera, which makes it home on the Simpson campus over the summer, with performances in Simpson's Blank Performing Arts Center.

Simpson for the murders of Nicole Brown Simpson and Ronald Goldman, in which Bugliosi asserted Simpson's guilt, and criticized the work of the district attorney, prosecutors, defense lawyers, and Judge Lance Ito ; he used these profiles to illustrate broader problems in American criminal justice, the media, and the political appointment of judges.

Simpson's hallmark legislation in the House of Representatives has been the Central Idaho Economic Development and Recreation Act ( CIEDRA ), which would create 312, 000 acres of wilderness in central Idaho, much of which is currently a wilderness study area.

Simpson's support for this form of energy plays a significant role in his membership of the United States House Appropriations Subcommittee on Energy and Water Development, which oversees the Idaho National Laboratory, a main site for nuclear and alternative energy research in the United States.

Brian Simpson's book on the subject mentions the Francis Spaight and also refers to a boat called Tiger on which a youth was cannibalized in 1766.

The reality show, which coincided with the release of the album, at first did not help Simpson's album sales, as it debuted at # 10 on the Billboard Charts and quickly fell down.

Zukertort died 20 June 1888, in London from a cerebral hemorrhage after playing a game in a tournament Simpson's Divan, which he was leading at the time.

Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases.

When the train line which terminated at Carlingford railway station, then known as Simpson's Railway, was completed in 1896, it too, became a major route for the area.

Gray was a co-writer and back-up singer on Jessica Simpson's album A Public Affair, which was released August 29, 2006.