NUMERICAL INTEGRATION 

Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral.

Trapezoidal Rule

In Calculus, “Trapezoidal Rule” is one of the important integration rules. The name trapezoidal is because when the area under the curve is evaluated, then the total area is divided into small trapezoids instead of rectangles. This rule is used for approximating the definite integrals where it uses the linear approximations of the functions.

The trapezoidal rule is mostly used in the numerical analysis process. To evaluate the definite integrals, we can also use Riemann Sums, where we use small rectangles to evaluate the area under the curve.

​

Trapezoidal Rule is a rule that evaluates the area under the curves by dividing the total area into smaller trapezoids rather than using rectangles. This integration works by approximating the region under the graph of a function as a trapezoid, and it calculates the area. This rule 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 a = x₀ < x₁< x₂< x₃<…..<x_n = b

Then the Trapezoidal Rule formula for area approximating the definite integral

​

​

is given by:

​

​

Where, x_i = a+iΔx

​

​

If n →∞, R.H.S of the expression approaches the definite integral

�

SIMPSON'S 1/3 RULE

​

Simpson's 1/3 rule is used to find the approximate value of a definite integral. Usually, we use the fundamental theorem of calculus to evaluate a definite integral. But sometimes, it is not possible to apply any of the integration techniques for the same.

​

​

Click to add text