**Simpson’s Rules may be used to find the areas and volumes of irregular figures. The rules are based on the assumption that the boundaries of such figures are curves which follow a definite mathematical law. When applied to ships they give a good approximation of areas and volumes.**

Simpson’s rule is a numerical method that approximates the value of a definite integral by using quadratic polynomials.

**The accuracy of the answers obtained will depend upon the spacing of the ordinates and upon how near the curve follows the law.**

**Simpson’s First Rule:-**

**This rule assumes that the curve is a parabola of the second order. A parabola of the second order is one whose equation, referred to co-ordinate axes, is of the form .**

y = (a_{0}+ a_{1}x + a_{2}x^{2})

** where a**_{0}, a_{1} and a_{2} are constants.

**where a**

_{0}, a_{1}and a_{2}are constants.**ln its simplest form, this rule states:-**

**ln its simplest form, this rule states:-**

**The area between any three consecutive ordinates is equal to the sum of the end ordinates,plus four times the middle ordinate, all multiplied by one third of the common interval.**

**Let the curve in Figure, be a parabola of the second order. Let y1, y2 and y3 be three ordinates equally spaced at ‘h’ units apart. The area of the elementary strip is y dx.Then the area enclosed by the curve and the axes of reference is given by:**

**Simpson’s Second Rule:-**

**Simpson’s Second Rule:-**

**This rule assumes that the equation of the curve is of the third order, i.e. of a curve whose equation, referred to the co-ordinate axes, is of the form**

y = a_{0}+ a_{1}x + a_{2}x^{2}+ a_{3}x^{3},

**where a0, a1, a2 and a3 are constants.**

**where a0, a1, a2 and a3 are constants.**

**The area between any four consecutive ordinates is equal to the sum of the end ordinates,plus threet imes eacho fthe middle ordinates, all multiplied by three-eighthso fthe common interval.**