Named after Thomas Simpson (1710-1761), Simpson’s Rule, in numerical integration, refers to the approximation for definite integrals. 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, 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 a quicker approximations 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.
In some other languages, such as German, the Simpson’s Rule hasn’t been named after Thomas Simpson, instead, it has been named after Johannes Kepler. This is because the latter derived it in 1615 when he first saw it being used for wine barrels.
FAQs on Simpson’s Rule
1. Find Out the Integral of the Function f(x) = 2x in the Interval (0, 2).
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,
\[\int_{a}^{b}\]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
2. Use the Simpson’s 1 / 3 Rule to Evaluate \[\int_{a}^{b}\] exdx
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,
\[\int_{a}^{b}\] f(x) dx = h / 3 [(y0 + yn) + 4 (1.1813 + 1.6487 + 2.3009) + 2 (1.39561 + 1.9477)]
Hence,
\[\int_{0}^{1}\] 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
3. What is Simpson’s Rule and where is it used?
The numerical method through the use of which a definite integral can be evaluated, is known as the Simpson’s Rule. And with the help of quadratic polynomials, Simpson’s Rule can be used to approximate the integrals. The fundamental theorem of calculus is used in case the definite integral is to be found. And, in that case, the antiderivative technique of integration is used. Therefore, it tends to give exact results when it comes to the approximation of integrals that are of polynomials up to a cubic degree.
4. What is the trapezoidal rule?
Also known as the trapezoid rule or the trapezium rule, the trapezoidal rule is a numerical method wherein the integrand is calculated approximately with the use of a second-order polynomial. It is considered to be a highly accurate method of approximating the area beneath a curve which is done by dividing the total area into small trapezoids instead of rectangles.
The formula for this rule is as follows:
\[\int_{a}^{b}\] f(x) dx = Δx / 2 [f(x_0) + 2f (x_1) + 2f (x_2) + …. 2f(x_{n-1}+f(x_n))],
where xi = a + i △ x
5. Discuss Simpson’s 1 / 3 rule and 3 / 8 along with their respective formulas.
Simpson’s 1 / 3 rule is actually said to be an extension of the trapezoidal rule and can give definitive results for finding the approximate polynomials.
The formula for Simpson’s 1 / 3 rule is as follows:
\[\int_{a}^{b}\] f(x) dx = h3[(y_0+ y_n) + 4n (y_1 + y_3 +....+y_n-1) + 2 (y_2 + y_4 + …. +yn_{-2})],
The Simpson’s 1 / 3 rule for integration:
By dividing a small interval a, b into two parts, it is possible to get a quicker approximation for definite integrals.
Simpson’s 3 / 8 rule is a great method that can be utilized for numerical integration as it is more efficient as well as more accurate as compared to the standard method. This is mainly due to the fact that it tends to use one more functional value and this numerical method is based on the cubic interpolation in its entirety, and not the quadratic interpolation. The formula used to represent Simpson’s 3 / 8 rule is as follows:
\[\int_{a}^{b}\] f(x) dx = 3h8[(y_0+ y_n) + (y_1 + y_2 + y_4 +....+y_{n-1}) + 2 (y_3 + y_6 + …. +yn_{-3})].
Simpson’s 3 / 8 rule is also commonly known as Simpson’s second rule of integration.
6. What is the difference between Simpson’s rule and the trapezoidal rule? Which one is better and why?
In mathematics, Simpson’s rule is defined as a numerical method that is used for the evaluation of a definite integral and for the approximation of the same. The trapezoidal rule, on the other hand, is a type of numerical method which uses the second-order polynomial to calculate the integral approximately. Although this rule has a much easier derivation compared to that of the other numerical method, and its conceptualization is also easier to comprehend, Simpson’s rule is better and more recommended. This is primarily due to the fact that Simpson’s rule is able to get a specific level of accuracy when it comes to the level of approximation, which the trapezoidal rule is not.
7. What is Simpson’s rule error?
Although it is better to use Simpson’s rule as it gives more accurate approximations compared to the other numerical methods, like the trapezoidal rule, there are still chances of an error or two occurring in this method because it gives an approximate value, not the exact value. And this is often defined when n = 2; -(1/ 90) (b - 1 / 2) 5f (4) (ξ).
Here, ξ is some number that tends to exist between a and b.
This points to the fact that the error tends to depend upon not only the fourth derivative of the actual function but also the distance between the points.