Simpson's Rule Calculator

This Simpson’s rule calculator helps to approximate the definite integrals step by step. Choose from Simpson's ⅓ or ⅜ Rule and get a precise estimation of the area under the curve. No matter, if you have a set of data points instead of a function, you can still use our Simpson's rule calculator for a table. Simply, choose your preferred method and let the calculator do the rest!

Note: Keep in mind the number of data points and the spacing between them is a crucial thing for the accuracy of the approximation. Meanwhile, increasing the number of data points enhances the accuracy of the results.

What is Simpson’s Rule in Calculus?

Simpson’s Rule is a numerical method that is used to approximate the definite integral. It indicates the area under the curve as a series of parabolas. This rule was named after Thomas Simpson. It's a way to approximate integrals rather than dealing with rectangles and performing the calculations.

In Simpson's Rule, parabolas are used to approximate each part of the curve. This method is more accurate as compared to other numerical methods.

The Basic Principle of Simpson’s Rule:

It states that:

“Given the 3 points, you can easily determine the quadratic for these points.”

There are 2 versions of Simpson’s rule ⅓ and ⅜. Simpsons 1/3 and 3/8 are two cases of Newton’s Cotes formulas. Simpson’s 3/8 rule requires the need for one more integral inside the integration range and gives lower error bounds.

The Formula for Simpson’s 1/3 Rule:

$\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(a) + 4 \sum_{\substack{i \text{ odd}}} f(x_{i}) + 2 \sum_{\substack{i \text{ even}}} f(x_{i}) + f(b) \right]$

$\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3} \left[ y_0 + 4(y_1 + y_3 + y_5 + ...) + 2(y_2 + y_4 + y_6 + ...) + y_n \right]$

Where:

$\ h= \frac{b - a}{n}$ (subinterval width)
n represents the number of subintervals(even)
f(a) and f(b) indicate the values of the function at the interval endpoints

Note: Remember that the interval of integration must be divided into an even number of subintervals and each subinterval must have the same width (h).

The Formula for Simpson’s 3/8 Rule:

$\int_{a}^{b} f(x) \, dx \approx \frac{3h}{8} \left[ f(a) + 3 \sum_{i=1}^{n-1} f(x_{i}) + 2 \sum_{i=3,6,9,...}^{n-1} f(x_{i}) + f(b) \right]$

The pattern of the coefficients in the Simpsons rule follows the pattern below:

$\ {\underbrace {1,4,2,4,2, \ldots ,4,2,4,1}_{{n + 1}\;\text{points}}.} $

Note: The interval of integration must be divided into a multiple of three subintervals and each subinterval must have the same width (h).

To simplify the application of Simpson's 1/3 rule and Simpson's 3/8 rule try our online Simpson's rule calculator. It uses the above formulas to efficiently compute definite integrals for a wide range of functions and table data, making the process both efficient and straightforward.

How To Implement Simpson’s Rule?

First of all, get the values of “a” and “b” from the interval [a,b]
Determine the number of subintervals (n)
Find the width of each subinterval using the formula “h = (b-a)/n”
Now, use the width of subintervals to divide the interval "[a, b]" in “n” number of subintervals
Put the values in Simpson’s rule (⅓ or ⅜ ) and get the result

Simpson’s Rule Example:

Approximate the area under the curve $\ y = 3^{x}$ between x = 0 and x = 1 by using the Simpsons rule with n = 2 subintervals.

Solution:

As the given curve is $\int\limits_{0}^{1} 3^{x}$, dx The Simpson’s rule formula is given as :

$$ \int\limits_{a}^{b} f(x)\, dx ≈ \dfrac{\Delta x}{3}(f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)) $$

Where the length of the interval is:

$$ \Delta x = \dfrac{b-a}{n} $$

As we have

a = 0,

b = 1,

n = 2

$$ \Delta x = \dfrac{1-0}{2} = 0.5 $$

Now, we have to divide the interval [0, 1] into 2 subintervals having length \Delta x = 0.5 for each endpoint:

a = 0, 0.5, 1 = b

Evaluating the function at these endpoints:

$$ f(x_{0}) = f(0) = 3^{0} = 1.0 $$

$$ 4f(x_{1}) = 4f(0.5) = (4*3)^{0.5} = 6.928203230275509 $$

$$ f(x_{2}) = f(1) = 3^{1} = 3.0 $$

Now add the values and multiply with $\dfrac{Δx}{3}$ = 0.25 0.25(1.0 + 6.928203230275509 + 3.0) = 1.821367205045918

The true solution for the integral is:

$$ \int\limits_{0.0}^{1.0} 3^{x}\, dx=2.0/log(3) $$

Finding the true solution for the integral can be time-consuming and prone to human errors. To deal with it, using a free Simpson’s rule calculator is a reliable choice. With it, you can get definite integrals precisely and instantly along with each step involved in the calculations.

What Are The Limitations of Simpson's Rule?

1. Oscillatory Functions:

The major drawback of using Simpson’s rule is that if we have a highly oscillatory function or lack derivatives at certain points, then this method is not suitable for finding accurate results.

2. Discontinuous Function:

When a function is discontinuous, points at which the derivative does not exist, or points with infinite derivatives, the accuracy in the approximation will reduce.

3. End Point Behavior:

If near the end point of the interval, functions have extreme behavior like rapid oscillations, or vertical asymptotes then the rule will not approximate the integral correctly.

How To Use The Simpson’s Rule Calculator?

Choose either Simpson's “⅓” or “⅜” rule
Now, select what you want to integrate a “function” or “table data” from the drop-down menu
Enter the necessary information in the input fields
Click on the “Calculate” button and determine the area under the curve

FAQ’s:

Why is Simpson’s Rule More Accurate?

It uses parabolas to approximate each part of the curve instead of a straight line, which is the most efficient method in numerical analysis. This method provides more accurate approximations of definite integrals than the other methods.

How To Improve The Accuracy of Simpson's Rule?

To increase the accuracy of Simpson’s rule increase the number of subintervals (n) that are used in the calculation.

What is The Order of The Error in The Simpson’s Rule?

As we know that the approximation for the function is quadratic, an order higher than the linear form, the error estimate of Simpson's rule is thus O ( h 4 ) or O ( h 4 f ‴ ) to be more specific.

References:

From the source of Wikipedia: Simpson's 1/3 rule, Composite Simpson's rule, Simpson's 3/8 rule, Composite Simpson's rule for irregularly spaced data.

From the source of inmath: Memory aid, proof for Simpson's Rule.