Trapezoidal Rule Formula
The trapezoidal rule formula is an integration rule used to calculate the area under a curve by dividing the curve into small trapezoids. The summation of all the areas of the small trapezoids will give the area under the curve. Let us learn this trapezoidal rule formula and a few solved examples in the upcoming sections.
What Is Trapezoidal Rule Formula?
Another useful integration rule is the Trapezoidal Rule. Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. This rule is used for approximating the definite integrals where it uses the linear approximations of the functions. The trapezoidal rule formula is:
Area=\(\dfrac{h}{2}[y_0+y_n+2(y_1+y_2+y_3+.....+y_{n1})]\)
where,
\(y_0\), \(y_1\),\(y_2\)…. are the values of function at x = 1, 2, 3….. respectively.
h = small interval (\(x_1 x_0\)
Solved Examples Using Trapezoidal Rule Formula

Example 1: Find the area under the curve using trapezoidal rule formula which passes through the following points:
x 0 0.5 1 1.5 y 5 6 9 11 Solution:
Given: y_{0}= 5
\(y_1\)= 6
\(y_2\)= 9
\(y_3\)= 11
h = (0.5  0) = (1  0.5) = (1.5  1) = 0.5
To Find: Area under the curveUsing Trapezoidal Rule Formula:
Area=\(\dfrac{h}{2}[y_0+y_n+2(y_1+y_2+y_3+.....+y_{n1})]\)
=\(\dfrac{0.5}{2}[5+11+2(6+9)]\)
=0.25[16+30]
=0.25[46]
=11.5
Answer: Area under the curve is 11.5 sq units.

Example 2 : Using Trapezoidal Rule Formula find the area under the curve y=x^{2} between x = 0 and x = 4 using the step size of 1.
Solution:
Given: y=x^{2}
h = 1
To Find: Area under the curveFind the values of ‘y’ for different values of ‘x’ by putting the value of ‘x’ in the equation y=x^{2}
X 0 1 2 3 4 y=x_{2} y_{0}= 0 y_{1}= 1 y_{2}= 4 y_{3}= 9 y_{4}= 16 Using Trapezoidal Rule Formula:
Area=\(\dfrac{h}{2}[y_0+y_n+2(y_1+y_2+y_3+.....+y_{n1})]\)
=\(\dfrac{1}{2}[0+16+2(1+4+9)]\)
=0.5[16+28]
=22
Answer: Area under the curve is 22 sq units.