Simpson’s Rule

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What is Simpson’s Rule?

In the simplest terms, it can be said that Simpson’s rule is a numerical method that can be used to evaluate a definite integral. In most cases, when we wish to find a definite integral, then that person uses the fundamental theorem of calculus. In that case, the antiderivative technique of integration is applied.


However, sometimes, it is very difficult to find the antiderivative of an integral. A good example of this is in the case of scientific experiments where the function has to be determined from the overall readings that were observed. Hence, it can be said that various numerical methods can be used to find the value of integrals in those cases.


But what exactly are those other numerical methods? According to experts, some of those numerical methods that can be used are midpoint rule, trapezoidal rule, left to right approximation by using Riemann sums, and Simpson formula. In this article, our focus will be on the Simpson formula. Readers will be able to understand the Simpson’s 1 / 3 rule, Simpson’s 3 / 8 rule, and Simpson’s rule integration.


Simpson’s Rule Formula

According to various sources, Simpson’s rule can be used for approximating the integrals. This is done by using quadratic polynomials. Here, the parabolic arcs are present in place of straight line segments that were used in the trapezoidal rule.


Simpson’s one-third rule can give definite results when it comes to finding the approximate polynomials. This can be done up to cubic degrees. It is important to remember that the trapezoidal formula would help in taking shape under a curve and find the area of those objects. However, if one wants to make those approximations better and more precise, then Simpson’s formula is the way to go.


One should also keep in mind that the simpson’s rule parabolas are used for finding parts of a curve. This means that according to the simpson meaning, the approximate area under the curve can be calculated by the following formula:


\[\int_{b}^{a}\] f (x) dx = h3 [(y0 + yn) + 4 (y1 + y3 + ….. + yn-1) + 2 (y2 + y4 + …. + yn-2)]


This formula is also known as the simpson’s 1 / 3 rule formula. Similarly, the simpson’s 3 / 8 rule formula is mentioned below.


\[\int_{b}^{a}\] f (x) dx = 3h8 [(y0 + yn) + (y1 + y2 + y4 + … + yn-1) + 2 (y3 + y6 + .. + yn-3)]


It is vital for our readers to note that the simpson’s 1 / 3 formula and simpson’s 3 / 8 rule formula is more accurate than any other methods of numerical approximations. The formula for n + 1 equally spaced subdivisions can also be given by the same method. However, in that case, n would be the even number, △x = (b - a) / n and xi = a + i△x


In this case, one must assume that we have f(x) = y. These are equally spaced between [a, b]  and if a = x0, x1 = x0 + h, x2 = x0 + 2h, ..., xn = x0 + nh. Here, h is the total difference between both the terms. We can also state that y0 = f(x0), y1 = f(x1), y2 = f(x2), …, yn = f(xn) are the analogous values of y with every value of x.


The Simpson’s 1/ 3rd Rule

Till now, readers must have understood an overview of the topic. Now, we will look at both the different types of rules and Simpson’s 1 / 3 rule example in more detail.


According to various sources, the Simpson’s 1 / 3 rule is an extension of the trapezoidal rule. For readers who are not familiar with the term, the trapezoidal rule is a numerical method in which the integrand is approximately calculated by using a second-order polynomial.


These facts indicate that if one uses Newton's divided difference polynomial, method of coefficients, and Lagrange polynomial, then one can derive this rule. Hence, the Simpson’s 1 / 3 rule can be defined by:


\[\int_{a}^{b}\]  f(x) dx = h / 3 [(y0 + yn) + 4(y1 + y3 + y5 + … + yn-1) + 2(y2 + y4 + y6 + … + yn-2)]


The Simpson’s 1 / 3 Rule for Integration

An individual can also get quicker approximation for definite integrals by dividing a small interval [a,b] into two parts. This means that after dividing the interval, one would get:


X0 = a, x1 = a + b, and x2 = b


This means that the approximation can be written as:


\[\int_{a}^{b}\] f(x) dx ≈ S2 = h / 3 [f(x0) + 4 f(x1) + f(x2)]


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


Here, h = (b-a) / 2


The Simpson’s 3 / 8 Rule

The Simpson’s 3 / 8 rule is another method that can be used for numerical integration. This numerical method is entirely based on the cubic interpolation instead of the quadratic interpolation. This rule can be represented by the formula that is mentioned below.


\[\int_{a}^{b}\] f(x) dx = 3h / 8 [(y0 + yn) + 3 (y1 + y2 + y4 + y5 + … + yn-1) + 2 (y3 + y6 + y9 + … + yn-3)]


This rule is more efficient and accurate than the standard method. This is because of the fact that this rule uses one more functional value. One should note that for the simpson’s 3 / 8 rule, there is also a composite simpson’s 3 / 8 rule. The latter is more similar to the generalized form. The simpson’s 3 / 8 rule is also known as the Simpson’s second rule of integration.


Simpson’s Rule Error

It is vital to note here that even though one gets a more accurate approximation by using Simpson’s rule method for definite integral calculation, errors still occur. This is defined when n = 2; -(1/ 90) (b - 1 / 2) 5f (4) (ξ)


Here, ξ is some number that exists between a and b.


The Graphical Representation of Simpson’s Rule

When it comes to mathematics, then no topic is complete without understanding the graphical representation of that topic. This is why we have attached an image below. This image showcases the graphical representation of Simpson's rule. Readers should carefully go over this image to understand this topic in a better light.

[Image will be Uploaded Soon]


Fun Facts about Simpson’s Rule Formula

Do you know that there is also an alternative and extended version of Simpson's rule? In that version, instead of applying Simpson’s rule to disjoint segments of the integral that have to be approximated, the Simpson’s rule is simply applied to the overlapping segments. This can be represented by the formula that is mentioned below.


∫ f(x) dx ≈ h / 48 [ 17 f(x0) + 59 f(z1) + 43 f (x2) + 49 f(z3) + 48 Σ f (x1) + 49 f(xn - 3) + 43 f(xn - 2) + 59 f(xn - 1) + 17 f(xn)]


This formula can be derived by combining the original composite Simpson’s rule with the formula that consists of using the Simpson’s 3 / 8 rule in the case of extreme subintervals and the standard 3 rule in the case of the remaining subintervals. After that, the mean of both the formulas is taken to obtain the final results.

FAQ (Frequently Asked Questions)

Question 1. Find Out the Integral of the Function f(x) = 2x in the Interval (0, 2).

Answer: It is given that A = 0 and B = 2. Let’s assume that n = 6

Hence, it can be said that

H = b - an = 2 - 0 x 6 = 13

X = 0 = a = 0

X1 = x0 + h = 0 = 13 = 13

X1 + h = 13 + 13 = 23

X2 + h = 23 + 13 = 33 = 1

X3 + h = 33 + 13 = 43

X4 + h = 43 + 13 = 53

X5 + h = 53 + 13 = 63 = 2

X6 + h = 63 + 13 = 1

X7 = b = 1

Y0 = f(0) = 2(0) = 0

Y1 = 2(13) = 23

Y2 = 2(23) = 43

Y3 = 2(33) = 2

Y4 = 2(43) = 83

Y5 = 2(53) = 103

Y6 = 2(63) = 4

Hence, according to the formula,

ba f(x) dx = h3 [(y0 + yn) + 4 (y1 + y3 + … + yn-1) + 2 (y2 + y4 + … + yn-2)]

f(x) dx = 133 [(0+4) + 4(23 + 2 + 103) + 2 (43 +83 + 4)]

= 1 / 9 [4 + 24 + 16]

= 44 / 9

= 4.89

Question 2. Use the Simpson’s 1 / 3 Rule to Evaluate ∫01exdx

Answer: To solve this question, let us divide the range (0, 1) into six equal parts by taking h = 1 / 6

This means that when x = 0, then y0 = e0 = 1

Now, when

X1 = x0 + h = 1 / 6, then y1 = e1 / 6 = 1.1813

X2 = x0 + 2h = 2 / 6 = 1 / 3, then y2 = e1 / 3 = 1.3956

X3 = x0 + 3h = 3 / 6 = 1 / 2, then y3 = e1 / 2 = 1.6487

X4 = x0 + 4h = 4 / 6 = 2 /3, then y4 = e2 / 3 = 1.9477

X5 = x0 + 5h = 5 / 6, then y5 = e5 / 6 = 2.3009

X6 = x0 + 6h = 6 /6 = 1, then y6 = e1 = 2.7182

Further, according to the Simpson’s 1 / 3 rule,

ab f(x) dx = h / 3 [(y0 + yn) + 4 (1.1813 + 1.6487 + 2.3009) + 2 (1.39561 + 1.9477)]

Hence,

01 exdx = 1 / 18 [(1 + 2.718) + 4(1.1813 + 1.6487 + 2.3009) + 2 (1.39561 + 1.9477)]

= 0.055 [ 3.7182 + 20.52422 + 6.6866]

= 1.71828