Trapezoidal Rule

In Calculus, an important integration rule is a “Trapezoidal Rule”, which is used for approximating the definite integrals where it uses the linear approximations of the functions. It is mostly used in the numerical analysis process. To evaluate the definite integrals, we can also use Riemann Sums. Let us discuss the Trapezoidal Rule formula for area, Trapezoidal Rule integration, and examples in this article.

Trapezoidal Rule Definition

Trapezoidal Rule

Trapezoidal Rule is a rule that evaluates the area under the curves by dividing the total area into smaller trapezoids rather than using rectangles. Trapezoidal Rule integration works by approximating the region under the graph of a function as a trapezoid, and it calculates the area. It takes the average of the left and the right sum. The Trapezoidal Rule does not give accurate value as Simpson’s Rule when the underlying function is smooth. It is because Simpson’s Rule uses the quadratic approximation instead of linear approximation. Both Simpson’s Rule and Trapezoidal Rule give the approximation value, but Simpson’s Rule results in even more accurate approximation value of the integrals.

Trapezoidal Rule Formula

Let f(x) be a continuous function on the interval [a, b]. Now divide the intervals [a, b] into n equal subintervals with each of width,

Δx = (b-a)/n, Such that x0 < x1< x2< x3<…..<xn = b,

Then the Trapezoidal Rule formula for area approximating the definite integral \(\int_{a}^{b}f(x)dx\) is given by:

\(\int_{a}^{b}f(x)dx\approx T_{n}=\frac{\bigtriangleup x}{2}[f(x_{0})+ 2f(x_{1})+2f(x_{2})+….2f(x_{n-1})+f(x_{n})]\)

Where, xi = a+iΔx

If n →∞, R.H.S of the expression approaches the definite integral \(\int_{a}^{b}f(x)dx\)

Trapezoidal Rule Examples

Go through the below given Trapezoidal Rule example.

Example 1:

Approximate the area under the curve y = f(x) between x =0 and x=8 using Trapezoidal Rule with n = 4 subintervals. A function f(x) is given in the table of values.

x 0 2 4 6 8
f(x) 3 7 11 9 3

Solution:

The Trapezoidal Rule formula for n= 4 subintervals is given as:

T4 =(Δx/2)[f(x0)+ 2f(x1)+ 2f(x2)+2f(x3) + f(x4)]

Here the subinterval width Δx = 2.

Now, substitute the values from the table, to find the approximate value of the area under the curve.

A≈ T4 =(2/2)[3+ 2(7)+ 2(11)+2(9) + 3]

A≈ T4 = 3 + 14 + 22+ 18+3 = 60

Therefore, the approximate value of area under the curve using Trapezoidal Rule is 60.

Example 2:

Approximate the area under the curve y = f(x) between x =-4 and x= 2 using Trapezoidal Rule with n = 6 subintervals. A function f(x) is given in the table of values.

x -4 -3 -2 -1 0 1 2
f(x) 0 4 5 3 10 11 2

Solution:

The Trapezoidal Rule formula for n= 6 subintervals is given as:

T6 =(Δx/2)[f(x0)+ 2f(x1)+ 2f(x2)+2f(x3) + 2f(x4)+2f(x5)+ f(x6)]

Here the subinterval width Δx = 1.

Now, substitute the values from the table, to find the approximate value of the area under the curve.

A≈ T6 =(1/2)[0+ 2(4)+ 2(5)+2(3) + 2(10)+2(11) +2]

A≈ T6 =(½) [ 8 + 10 + 6+ 20 +22 +2 ] = 68/2 = 34

Therefore, the approximate value of area under the curve using Trapezoidal Rule is 34.

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