Simpson’s rule for approximating definite integrals was first published by Thomas Simpson in 1743. Although named after him, Thomas Simpson was not actually the one who first discovered it. Bonaventura Cavalieri discovered an early version of this rule in 1639 and James Gregory published this variation in1668along with some other numerical methods for approximating definite integrals.
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. In this article, we’ll look at the Simpson’s rule formula, the 1/3 rule, the 3/8 rule, and some examples in this regard.
What is the order of Simpson’s 3/8 rule?
Simpson’s 3/8 approximates the value of a definite integral and is given by the formula: 3/8 * Delta x f(x_0) + 3f(x_1) + 3f(x_2) + 2f(x_3) + … + 3f(x_{n-1}) + f(x_n) and it approximates the integral from a to b of f(x) dx. Here, n must be a multiple of three and Delta x is given by (b-a)/n. The order of the coefficients follows the pattern of 1, 3, 3, 2, 3, 3, 2, …, 3, 3, 1.
What is Simpson’s 1/3rd rule?
Simpson’s 1/3 rule is an approximation for definite integrals. It states that for an integral from a to b of f(x) dx, it is approximately Delta x/ 3 f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + … + 4f(x_{n-1}) + f(x_n). Here, n must be even and Delta x = (b-a)/n. This formula is exact for polynomials, f(x), up to degree two.
What is Simpson’s 1/3rd rule?
Simpson’s 1/3 rule is an approximation for definite integrals. It states that for an integral from a to b of f(x) dx, it is approximately Delta x/ 3 f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + … + 4f(x_{n-1}) + f(x_n). Here, n must be even and Delta x = (b-a)/n. This formula is exact for polynomials, f(x), up to degree two.
How is Simpson’s rule calculated?
Simpson’s rule is calculated by first identifying n, the number of divisions of the integration region and calculating Delta x. Then, identify the different input values that the formula will use by using Delta x and the upper and lower limits of the integral.
Input these values into the formula while controlling for the coefficient pattern. Finally, simply calculate the output values of the function substitute them into the formula; then simplify to get the final approximation.
Simpson’s 1/3 Rule vs Simpson’s 3/8 Rule: Key Differences
| Points of Comparison | Simpson’s 1/3 rule | Simpson’s 3/8 rule |
|---|---|---|
| Use | It approximates function y = f(x) by a parabola i.e. by 2nd order polynomial. | It approximates the function y = f(x) by a parabola i.e. by 3rd order polynomial. |
| Chance of Error | In this, the chances of error are more than Simpson’s 3/8 rule. | In this, the chances of error are less. |
| Integral Function | The integral function can be calculated as = h/3 [(sum of 1st and last ordinates) + 4 (sum of odd ordinates) + 2 (sum of even ordinates)]. | The integral function can be calculated as = 3h/8 [(sum of 1st and last ordinates) + 2 (sum of multiple of 3 ordinates) + 3 (sum of remaining ordinates)]. |
| Estimation | Estimation in truncation error in Simpson ‘s 1/3 rule is E< -nh5/180 y iv(x) where h = (b-a)/n | Estimation in truncation error in Simpson ‘s 3/8 rule is E< -nh5/90 y iv(x) where h = (b-a)/n |
| Application | This rule is applied where N is an even number. | This rule is applied where N is a multiple of 3. |
Applications of Simpson’s Rule
- Simpson’s rule finds the values of the definite integral using quadratic or cubic curves, depending on the precision required.
- Determining static and dynamic reaction forces on surfaces and volumes.
- When developing a new maritime vessel, resolving buoyancy and stability issues.
- Calculating average power across an infinite number of voltage and current cycles.
Limitations of Simpson’s rule
- 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).
- Integrals allow you to get exact answers in terms of fundamental constants, which Simpson’s method does not allow.
- To get a good approximation to the real integral, it is necessary (often) to use a large number of ordinates.
Key Takeaways
- Simpson’s 1/3rd rule is an extension of the trapezoidal rule in which the integrand is approximated by a second-order polynomial.
- The 1/3-rule uses a quadratic approximation to the curve, and requires number of sample points to be a multiple of 2.
- The 3/8-rule uses a cubic approximation to the curve, and requires number of sample points to be a multiple of 3.