To the sum of the first and last ordinates, add twice the sum of the intermediate ordinates. The total sum thus obtained is multiplied by the common distance between the ordinates. One-half of this product gives the required area. This rule of finding the area is called

Concept:

i) Mid-Ordinate Rule

Area of plot = h1 × d + h2 × d + … + hn × d = d (h1 + h2 + … hn)

∴ Area = common distance × sum of mid-ordinates

ii) Average-Ordinate Rule

$\text{Area}=\frac{{\text{O}}_1+{\text{O}}_2+\dots +{\text{O}}_{\text{n}}}{{\text{O}}_{\text{n}+1}}\times \text{l}$

$\text{i}.\text{e}.\text{ Area}=\frac{\text{Sum }\text{ of }\text{ ordinates}}{\text{No}.\text{ of }\text{ ordinates}}\times \text{length }\text{ of }\text{ base }\text{ line}$

(iii) Trapezoidal Rule

$\text{Total }area=\frac{\text{d}}{2}\{{\text{O}}_1+2{\text{O}}_1+2{\text{O}}_2+\dots +2{\text{O}}_{\text{n}-1}+{\text{O}}_{\text{n}}\}$

$\text{Total Area}=\frac{\text{Common distance}}{2}\times \{(1\text{st ordinate}+\text{last ordinate})+2\left(\text{sum of other ordinate}\right))\}$

(iv) Simpson’s Rule

$\text{Total }area=\frac{\text{d}}{3}({\text{O}}_1+4{\text{O}}_2+2{\text{O}}_3+4{\text{O}}_4+\dots +{\text{O}}_{\text{n}})$

$\text{Total Area}=\frac{\text{Common Distance}}{3}\times \{(1\text{st ordinate}+\text{last ordinate})+4\left(\text{sum of even ordinates}\right)+2\left(\text{sum of remaining odd ordinates}\right)\}$