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 o ...

, to calculate the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

s and volumes of irregular figures. This is an application of Simpson's rule for finding the values of an

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

, 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

lifeboat Lifeboat may refer to: Rescue vessels * Lifeboat (shipboard), a small craft aboard a ship to allow for emergency escape * Lifeboat (rescue), a boat designed for sea rescues * Airborne lifeboat, an air-dropped boat used to save downed airmen A ...

s, and by surveyors to calculate the volume of sludge 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