Simpson’s Rule Real Life Applications

This approach is named after Thomas Simpson (1710–1761), an English mathematician.

One of the numerical approaches for evaluating the definite integral is Simpson’s rule. 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 utilized to approximate the integral. Trapezoidal rule, midpoint rule, and left or right approximation using Riemann sums are some of the other numerical methods used. We’ll go through the Simpson’s rule formula, the 1/3 rule, the 3/8 rule, and some examples in this section.

Definition

Simpson’s Rule is based on the idea that we can find the equation of a quadratic through three points if we have three points.

To get a rough estimate of the definite integral b∫a f(x) dx We divide the interval [a, b] into an even number of n subintervals, each of width [a, b] using Simpson’s Rule.

∆X = (b-a)/n

We consider a quadratic function for each pair of subsequent subintervals [ Xᵢ₋₁ , Xᵢ ] , [ Xᵢ , Xᵢ ₊₁ ] ,

We choose a quadratic function y= ax²+bx+c that goes across the points as an example.

( Xᵢ₋₁ , ƒ ( Xᵢ₋₁ ) ) , ( xᵢ , ƒ ( xᵢ ) ) , ( xᵢ₊₁ , f ( xᵢ₊₁ ) ) .

If the function f(x) is turned on (a,b) all the time,

b∫a f(x) dx ≈ ∆X /3 [ ƒ ( x₀ ) + 4f ( x₁ ) + 2ƒ ( x₂ ) + 4f ( x₃ ) + 2ƒ ( x₄ ) + … + 4ƒ ( xₙ₋₁ ) + f ( xₙ ) ] . 

In Simpson’s Rule, the coefficients follow this pattern:

1 , 4 , 2 , 4 , 2 , … , 4 , 2 , 4 , 1 . = n + 1 points.

Simpson’s 1/3 Rule Derivation

Let us derive Simpson’s 1/3 rule, which involves dividing the area under the curve f(x) into parabolas to approximate the value of the definite integral ba f(x) dx. Divide the interval [a, b] into n subintervals for this [x₀, x₁], [x₁, x₂], [x₂, x₃], …, [xₙ₋₂, xₙ₋₁], [xₙ₋₁, xₙ] each of width ‘h’, where x₀ = a and xₙ = b.

           The interval (a,b) is divided into subintervals

            Of width h

Let us now approximate the area under the curve by assuming that every three consecutive points lie on a parabola. By sketching a parabola between the points x₀, x₁, and x₂, we may approximate the area under the curve between x₀ and x₂. Naturally, all three may not appear on the same parabola. However, let us attempt to draw a parabola through these three locations

Let’s make the y-axis of this parabola symmetric. After that, it looks like this:

Let’s pretend that the parabola’s equation is y = ax² + bx + c. The definite integral is then used to approximate the area between x₀ and x₂:

Area between x₀ and x₂ ≈ ₋ₕ∫ʰ (ax² + bx + c) dx

= (ax³/3 + bx²/2 + cx) ₋ₕ|ʰ

= (2ah³/3 + 0 + 2ch)

= h/3 (2ah² + 6c) … (1)

Let us have another observation from the above figure.

F(x₀) = a(-h)² + b(-h) + c = ah² – bh + c

F(x₁) = a(0)² + b(0) + c = c

F(x₂) = a(h)² + b(h) + c = ah² + bh + c

Now, f(x₀) + 4f(x₁) + f(x₂) = (ah² – bh + c) + 4c + (ah² + bh + c) = 2ah² + 6c.

Substitute this in (1):

Area between x₀ and x₂ ≈ h/3 (f(x₀) + 4f(x₁) + f(x₂))

Similarly, we can see that:

Area between x₂ and x₄ ≈ h/3 (f(x₂) + 4f(x₃) + f(x₄))

Calculating the other areas in a similar way, we get

b∫ₐ f(x) dx

= h/3 (f(x₀) + 4f(x₁) + f(x₂))+ h/3 (f(x₂) + 4f(x₃) + f(x₄))+ …+ h/3 (f(xₙ₋₂) + 4f(xₙ₋₁) + f(xₙ))

≈ (h/3) [f(x₀)+4 f(x₁)+2 f(x₂)+ … +2 f(xₙ₋₂)+4 f(xₙ₋₁)+f(xₙ)]

The terms that are similar have been combined here.

As a result, we have the formula for Simpson’s rule.

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: From the interval [a, b], get the values of ‘a’ and ‘b,’ as well as the value of ‘n,’ which is 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)]

Important Notes on Simpson’s Rule

• We always divide the interval into an even number of subintervals when using Simpson’s rule. In other words, ‘n’ must always be even.
• The breadth of the subintervals must be equal.
• According to Simpson’s 1/3 rule:

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)]

Applications of Simpson’s Rule

One of the numerical approaches for evaluating the definite integral is Simpson’s rule. 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 utilized to approximate the integral. Trapezoidal rule, midpoint rule, and left or right approximation using Riemann sums are some of the other numerical methods used.

Simpson’s rule finds the values of the definite integral using quadratic or cubic curves, depending on the precision required. Cubic curves provide more precision.

1. Determining static and dynamic reaction forces on surfaces and volumes.

2. When developing a new maritime vessel, resolving buoyancy and stability issues.

3. Calculating average power across an infinite number of voltage and current cycles.

Limitations of Simpson’s rule

1. It is obviously inaccurate, i.e. there will always be a difference between it and the actual integral (except in some cases, such as the area under straight lines).
2. Integrals allow you to get exact answers in terms of fundamental constants, which Simpson’s method does not allow.
3. To get a good approximation to the real integral, it is necessary (often) to use a large number of ordinates.

Conclusion

If f is a polynomial of up to third degree, the rule’s approximation equality becomes precise.

The composite Simpson’s rule is obtained by applying the 1/3 rule to n equal subdivisions of the integration range [a, b]. Weights of 4/3 and 2/3 are alternated for points within the integration range.

Simpson’s approach, which approximates functions with quadratic polynomials, actually produces accurate results when approximating integrals of functions.