Overview of the Simpson’s Rule in Calculus

Simpson’s rule is one of the numerical methods for assessing the definite integral. To get the definite integral, we usually employ the fundamental theorem of calculus, which requires us to use antiderivative integration techniques. However, in other cases, such as in Scientific Experiments, where the function must be calculated from observed data, finding the antiderivative of an integral is difficult. In such situations, numerical approaches are utilised to approximate the integral. Other numerical methods include the trapezoidal rule, midpoint rule, and left or right approximation using Riemann sums. 

Simpson’s rule: 

By approximating the area under the graph of the function f, Simpson’s technique is used to find the value of a definite integral (that is, of the type b∫a f(x) dx). We calculate the area under a curve (a definite integral) using the Riemann sum by dividing the area under the curve into rectangles, but we evaluate the area under a curve using Simpson’s formula by dividing the whole area into parabolas. Simpson’s 1/3 rule (pronounced Simpson’s one-third rule) is another name for Simpson’s rule. 

Simpson’s rule formula: 

To estimate an integral, we can use Riemann’s left sum, Riemann’s right sum, midpoint rule, trapezoidal rule, Simpson’s 1/3 rule, and other numerical methods. Simpson’s approach, however, provides a more precise approximation of a definite integral. The Simpson’s rule formula, f(x) = y, which is evenly spread between [a,b], is: 

b∫a f(x) d x ≈ (h/3) [f(x0)+4 f(x1)+2 f(x2)+ … +2 f(xn-2)+4 f(xn-1)+f(xn)] 

Here, 

• n is an even number that represents the number of subintervals that should be divided into the interval [a, b] (In most cases, n is mentioned in the problem). 
•  x0 = a; xn = b 
• h = [ (b – a) / n] 
• The ends of the n subintervals are x0, x1,…, xn. 

Simpson’s ⅓ rule: 

Simpson’s 1/3rd rule, in which the integrand is approximated by a second-order polynomial, is a variation on the trapezoidal rule that is more general. The Simpson rule can be determined in a variety of ways, including utilising Newton’s divided difference polynomial, Lagrange polynomial, and the coefficients technique. The 1/3 rule, as defined by Simpson, is as follows: 

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

This guideline is known as Simpson’s One-third Rule. 

Simpson’s ⅓ rule for integration: 

Using a small interval [a, b] to divide into two equal halves, we can obtain a fast approximation for definite integrals. We get, as a result of splitting the interval: 

x0= a, x1= a + b, x2 = b 

As a result, the approximation can be written as; 

∫ab f(x) dx ≈ S2 = h/3[f(x0) + 4f(x1) + f(x2)]

S2 = h/3 [f(a) + 4 f((a+b)/2) + f(b)] 

Where, h = (b – a)/2 

For integration, this is Simpson’s ⅓ rule. 

Simpson’s ⅜ rule: 

The “Simpson’s 3/8 rule” is another way of numerical integration. Rather than quadratic interpolation, it is entirely based on cubic interpolation. The Simpson 3/8 or three-eight rule is defined as follows: 

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

This rule is more accurate than the standard method since it uses an extra functional value. The composite Simpson’s 3/8 rule, which is comparable to the generalised form, also exists for the 3/8 rule. The 3/8 rule is Simpson’s second rule of integration. 

Simpson’s Rule Error: 

Even though we achieve a more precise approximation for the definite integral using Simpson’s rule method, we still receive the error that is defined for n = 2; 

-(1/90)[(b-a)/2]5 f(4) (ξ) 

Between a and b, there is some number ξ. 

How to apply Simpson’s rule: 

The 1/3 rule of Simpson provides a more precise approximation. The next steps will show you how to use Simpson’s rule to approximate the integral b∫a f(x) dx. 

• Step 1: Determine the values of ‘a’ and ‘b’ in the interval [a, b], as well as the value of ‘n,’ the number of subintervals. 
• Step 2: Calculate the width of each subinterval using the formula h = (b – a)/n. 
• Step 3: Using the interval width ‘h,’ divide the interval [a, b] into ‘n’ subintervals [x0, x1], [x1, x2], [x2, x3],…, [xn-2, xn-1], [xn-1, xn]. 
• Step 4: Simplify Simpson’s rule formula by substituting all of these values. 

b∫ₐ f(x) dx ≈ (h/3) [f(x0)+4 f(x1)+2 f(x2)+ … +2 f(xn-2)+4 f(xn-1)+f(xn)] 

Simpson’s rule example: 

1. By Simpson’s ⅓ rule, evaluate ∫01exdx. 

Solution. Using h = 1/6, divide the range [0, 1] into six equal sections. 

If x0 = 0 then y0 = e0 = 1.

If x1 = x0 + h = ⅙, then y1 = e1/6 = 1.1813

If x2 = x0 + 2h = 2/6 = 1/3 then, y2 = e1/3 = 1.3956

If x3 = x0 + 3h = 3/6 = ½ then y3 = e1/2= 1.6487

If x4 = x0 + 4h = 4/6 ⅔ then y4 = e2/3 = 1.9477

If x5 = x0 + 5h = ⅚ then y5 = e5/6 = 2.3009

If x6 = x0 + 6h = 6/6 = 1 then y6 = e1 = 2.7182 

We are aware of Simpson’s ⅓ rule; 

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

As a result, 

∫01exdx = (1/18) [(1 + 2.7182) + 4(1.1813 + 1.6487 + 2.3009) + 2(1.39561 + 1.9477)]

=  (1/18)[3.7182 + 20.5236 + 6.68662]

= 1.7182 (approx.) 

Conclusion: 

In order to calculate the value of a definite integral, Simpson’s Rule is used to approximate the area under the graph of the function f. The area under a curve is calculated by splitting it into parabolas in Simpson’s method, but the area under a curve is calculated by dividing it into rectangles in Riemann Sum (a definite integral).