The Newton-Cotes integration formulas are the most commonly used numerical integration methods in numerical analysis. The strategy involved in all these Newton cotes formulas is replacing a complicated function with an approximating function which helps us perform the integration easily. Thus, Newton cote integration formulas play an important role in solving numerical integration problems. These formulas help in deriving the simple form of given expressions in integration. Let’s learn different types of Newton cotes integration formulas here in this article.

What are Newton Cotes Integration Formulas?

In numerical analysis, the Newton–Cotes formulas, or Newton Cotes rules, are also called the Newton–Cotes quadrature rules, which comprises a group of formulas for numerical integration based on estimating the integrand at equally spaced points. Also, these formulas or rules were named after Isaac Newton and Roger Cotes. Below are a few important Newton cotes integration rules.

• Trapezoidal Rule
• Simpson’s Rules
• Simpson’s 1/3 Rule
• Simpson’s 3/8 Rule

Before learning about all these rules, let’s recall what is numerical integration.

Numerical integration:

Numerical integration is the approximate estimation of integral employing numerical methods. Sometimes, the numerical estimation of an integral is called quadrature.

Trapezoidal Rule

The trapezoidal rule uses the first-degree polynomial to replace the integrand, i.e., the function to be integrated.

Consider the integration of f(x) over the interval [a, b]

I = ∫_a^b f(x) = ∫_a^b f₁(x)

Here, f1(x) can be written as:

f₁(x) = a + [{f(b) – f(a)}/ (b – a)] (x – a)

Therefore,

I = (b – a) [f(a) + (b)]/2

The only efficient way to enhance the precision of the trapezoidal rule is to divide the integration interval [a, b] into several segments and apply the technique to each of these segments. Then, by adding the areas of all these parts, we can get the integration of the entire function. Mathematically, we can define this technique as follows.

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

Δx = h = (b – a)/n, such that a = x₀ < x₁< x₂< x₃<…..<x_n = b and a = x₀, b = x_n.

Then the Trapezoidal Rule formula for area approximating the definite integral ∫_a^b f(x) is given by:

Here,

x_i = a + iΔx = a + ih

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

This trapezoidal method of converting the interval of integration into segments is called the Multiple trapezoidal rule.

Click here to learn more about the trapezoidal rule of integration.

Simpson’s Rules

The most efficient way to get a better accurate estimation of an integral is to operate a higher-order polynomial to join the points. This way of estimating the integrals is known as Simpson’s Rule.

Simpson’s formula for n+1 equally spaced subdivision is given by;

Here,

n = even number

△x = h = (b – a)/n

x_i = a + i△x = a + ih

If the function y = f(x) is equally spaced between [a, b] and if a = x₀, x₁ = x₀ + h, x₂ = x0 + 2h …., x_n = x₀ + nh, where h is the difference between the terms, then the analogous values of y with each value of x will be y₀ = f(x₀), y₁ = f(x₁), y₂ = f(x₂),……, y_n = f(x_n).

Simpson’s 1/3 Rule

Simpson’s one-third rule or Simpson’s ⅓ rule is generally considered as an extension of the trapezoidal rule in which a second-order polynomial approximates the integrand. Simpson’s rule can be derived from the various ways using Newton’s divided difference polynomial, Lagrange polynomial, and the coefficients method. Simpson’s 1/3 rule is defined by:

∫_a^b f(x) dx = h/3 [(y₀ + y_n) + 4(y₁ + y₃ + y₅ + …. + y_n-1) + 2(y₂ + y₄ + y₆ + ….. + y_n-2)]

This rule is known as Simpson’s One-third rule.

Simpson’s 3/8 Rule

Simpson’s 3/8 rule is based on cubic rather than quadratic interpolation. Simpson’s 3/8 rule or three-eight rule is given by the formula:

∫_a^b f(x) dx = 3h/8 [(y₀ + y_n) + 3(y₁ + y₂ + y₄ + y₅ + …. + y_n-1) + 2(y₃ + y₆ + y₉ + ….. + y_n-3)]

The 3/8 rule is known as Simpson’s second rule of integration.

Get more information about Simpson’s rule here

Apart from the above formulas, we can also define some special formulas such as open and closed newton cotes.

Closed and Open Newton Cotes Formulas

Let us assume that the value of f(x) defined on the interval [a, b] is known at n + 1 equally spaced points, i.e., x₀, x₁, x₂,…, x_n such that a ≤ x₀ < x₁ < x₂ < …. < x_n ≤ b. Thus, there will be two classes of Newton cotes quadrature which are called closed and open Newton cotes. Closed Newton cotes occur when a = x₀ and b = x_n, i.e., they use the function values at the endpoints of the interval. Also, open cotes occur when a < x₀ and x_n < b, which means they do not use the values of the function at the endpoints.

Open and closed Newton cotes formulas using (n +1) points can be defined as follows.

Here,

For an open newton cotes formula, x_i = a + i△x = a + ih such that h = (b – a)/n.

For a closed newton cotes formula, x_i = a + (i + 1)△x = a + (i + 1)h such that h = (b – a)/(n + 2).

Also, h represents the step size and w_i represents the weights.

Open Newton Cotes Formulas

Some open Newton cotes formulas for integration are tabulated below.

Closed Newton Cotes Formulas

A few closed Newton cotes formulas for integration are given in the table below.

Solved Examples

Example 1:

What is the Newton Cotes formula for Trapezoidal rule?

Solution:

We can write the closed Newton cotes formula for Trapezoidal rule as:

(h/2)(f₀ + f₁)

Here,

h = (b – a)

Example 2:

Write the Open Newton Cotes integration for n = 2.

Solution:

For n = 2, the Open Newton cotes formula for integration is given by:

(4/3)h (2f₁ – f₂ + 2f₃)

Here,

n = 2

h = (b – a)/4