Simpson's rules are a set of rules used in ship stability and

naval architecture Naval architecture, or naval engineering, is an engineering discipline incorporating elements of mechanical, electrical, electronic, software and safety engineering as applied to the engineering design process, shipbuilding, maintenance, and op ...

, to calculate the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

s and

volumes Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The ...

of irregular figures. This is an application of

Simpson's rule In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, ...

for finding the values of an

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

, here interpreted as the area under a curve.

Simpson's 1st rule

Also known as the 1–4–1 rule (after the multipliers used ). :

\text = \frac( a + 4b + c).

Simpson's 2nd rule

Also known as the 1–3–3–1 rule, Simpson's second rule is a simplified version of Simpson's 3/8 rule. :

\text = \frac 8 (a + 3b + 3c + d).

Simpson's 3rd rule

Also known as the 5–8–1 rule, SImpson's third rule is used to find the area between two consecutive ordinates when three consecutive ordinates are known. :

\text = \frac h  (5a + 8b - c ).

This estimates the area in the left half of the figure for Simpson's 1st Rule while using all three pieces of data.

Use of Simpsons rules

Simpson's rules are used to calculate the volume of lifeboats, and by surveyors to calculate the volume of

sludge Sludge (possibly , or some dialect related to slush) is a semi-solid slurry that can be produced from a range of industrial processes, from water treatment, wastewater treatment or on-site sanitation systems. It can be produced as a settled sus ...

in a ship's oil tanks. For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used. Simpson's rules are used by a ship's officers to check that the area under the ship's GZ curve complies with IMO stability criteria.

References

{{Reflist Naval architecture