Trapezoidal Rule
To find the area of the trapezium as shown in the figure, the base is divided into equal intervals of width
. The ordinates
are accurately measured. The approximation used in this rule is to assume that each strip is equal to the area of a trapezium. Therefore:
Area of a trapezium (sum of parallel sides)
(perpendicular distance between the parallel sides)

Hence, the first strip's approximate area is
For the second strip, the approximate area is , and so on.
Therefore, the approximate area of
Area
width of interval
sum of first and last ordinate
+ sum of remaining ordinates
Example:
Find the area of a cross-section of a river along a line where the depths at equal intervals of m are noted as
m, respectively.
Solution:
Width of each strip, m
Ordinates are
Since,
Area
width of interval
sum of first and last ordinate
+ sum of remaining ordinates
square meters.
Example:
Apply the trapezoidal rule to find the area of a plot of land having the following dimensions:
Ordinates: and
m
Common distance: m
Solution:
Given that m,
m,
m,
m,
m respectively and
m
by the trapezoidal rule:
Area
square meters.