Trapezoid Rule Formula with Solved Examples

Trapezoid Rule Formula 

The trapezoid rule is usually an integration of rules that are used to calculate the area under a curve just by dividing the curve into smaller trapezoids. The sum of all small trapezoids will give us the area under the curve. According to the trapezoidal rule, we can evaluate the area under the curve by dividing the total area of a small trapezoid rather than any rectangle. 

Now let’s apply the trapezoid Rule Formula to solve the definite integral by calculating the area under the curve by dividing the total area into small trapezoids rather than rectangles. This rule is usually used to approximate the definite integral where it requires a linear approximation of the function. The trapezoid Rule averagely requires the left and right sum. 

Let y = f(x) continues on a,b. Now let’s divide (a,b) by n equal sub intervals each of its width 

h = (b-a)/n 

So, X1<X2<X3<……Xn 

Area = (h/2) {y1 +2 (y1+y2+y3….Yn-1) Yn 

Solved Examples

Example 1 

Find out the area under the curve with help of the Trapezoid Rule Formula that passes through the following points

Given, 

Y0 – 5, Y1 – 6, Y2 – 9, Y3 – 11 

h – 0.5-0 = 0.5 

Area of Trapezoid Formula = (h/2) {y0 + yn +2 (y1+y2+y3….Yn-1)} 

                          = 0.5/2 {5+11+2 (6+9)} 

                         = 11.2 sq units 

Thus, the area under the curve is 11.2 sq units. 

Example 2 

With the help of the Trapezoidal rule, the formula finds the area under the curve y = X sq between x = 0 and x = 4 using the step size of 1 

Given, y = x2 

h = 1 

Let’s find out the value of y with help of this y = x2 

Using trapezoid Rule

Area of under the curve= (h/2) {y0 + Yn +2 (y1+y2+y3….Yn-1)} 

          = 1/2 {0+16+2(1+4+9)} 

           = 22 

Therefore, an area under the curve is 22 sq units.