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Math Calculators ▶ Simpson's Rule Calculator

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**Table of Content**

An online Simpson’s rule calculator is programmed to approximate the definite integral by determining the area under a parabola. You can work for the equation of the quadratic by using the Simpson calculator.

For a better understanding of the concept of Simpson’s rule, give it a proper read.

In mathematics, **a numerical approximation of a definite integral by using quadratic functions is known as Simpson’s rule**. Rather than calculating the area of narrow rectangles, an online Simpsons rule calculator is the best option to evaluate the area under the curve as a whole.

**Basic principle of Simpson’s rule:**

It states that:

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

Let us suppose we are given the definite integral as follows:

$$ \int\limits_a^b {f\left( x \right)dx} $$

Now, if we want to get the suitable approach of the above integral, we need to make partition of the interval [a, b] into subintervals of even numbers n.

The width of each subinterval is given by:

$$ {\Delta x = \frac{{b – a}}{n}.} $$

If there are three points as:

$$ \left( {{x_{i – 1}},f\left( {{x_{i – 1}}} \right)} \right) $$

We assume that the quadratic y = a{x^2} + bx + c passes from all the above three points mentioned and is defined for each pair of consecutive subintervals

$$ \left({{x_{i – 1}},{x_i}} \right), \left( {{x_i},{x_{i + 1}}} \right) $$

If the function f(x) is continuous on the interval [a, b] then we have the simpson’s rule equation as follows:

$$ {\int\limits_a^b {f\left( x \right)dx} }\approx{ {\frac{{\Delta x}}{3}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) }\right.}+{\left.{ 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) }\right.}+{\left.{ 2f\left( {{x_4}} \right) + \cdots }\right.}+{\left.{ 4f\left( {{x_{n – 1}}} \right) + f\left( {{x_n}} \right)} \right].} $$

Because of the factor of 1 / 3 in the formula, it is also known as Simpsons 1 / 3 rule. Furthermore, a free Simpsons 1/3 rule calculator is one of the best ways to solve a definite integral precisely.

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

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

Our free online Simpson’s rule calculator works on the above formula to evaluate the definite integral.

Before we start applying Simpson’s rule we must know the proper use of it. That is why we have to solve a couple of problems for having a hands-on grip.

**Problem # 01:**

Use simpsons rule with n = 3 to approximate the integral below:

$$ \int\limits_0^6 {\sqrt x dx}. $$

**Solution:**

The given integral is:

$$ \int\limits_{0}^{6} \sqrt{x}\, dx $$

Using Simpson’s rule formula:

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

As we know that the width of each interval is given as:

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

As we have:

**a = 0, b = 6, n = 4**

$$ \Delta x = \dfrac{6-0}{4} = 1.5 $$

Now, you need to divide the interval [0, 6] having length of the arc to be Δx=1.5 for the following endpoints:

**a = 0, 1.5, 3, 4.5, 6**

Now, we have to evaluate the function at these endpoints:

$$ f(x_{0}) = f(0) = \sqrt{0} = 0.0 $$

$$ 4f(x_{1}) = 4f(1.5) = 4\sqrt{1.5} = 4.898979485566356 $$

$$ 2f(x_{2}) = 2f(3) = 2\sqrt{3} = 3.4641016151377544 $$

$$ 4f(x_{3}) = 4f(4.5) = 4\sqrt{4.5} = 8.48528137423857 $$

$$ f(x_{4}) = f(6) = \sqrt{6} = 2.449489742783178 $$

Add all the values and multiply with \dfrac{Δx}{3} = 0.75

**0.75(0.0 + 4.898979485566356 + 3.4641016151377544 + 8.48528137423857 + 2.449489742783178) = 9.64892610886293**

The actual solution of the integral is given as:

$$ \int\limits_{0}^{6} \sqrt{x}\, dx = -4 \ sqrt{6} $$ (click integral calculator for calculations)

$$ \int\limits_{0}^{6} \sqrt{x}\, dx = 9.7979589711 $$

Hence, the error involved in the approximation of the integral is as follows:

$$ {\left| \varepsilon \right| = \left| {\frac{{9.7979589711 – 9.64892610886293}}{{9.7979589711}}} \right| }\approx{ 0.015 }={ 1.5\%} $$

**= 0.01521**

**= 1.521%**

You can also correspond to free online Simpson’s rule calculator to find the error more precisely.

**Problem # 02:**

Approximate the area under the curve y = 3^{x} between x = 0 and x = 1 by using 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:

$$ \int\limits_{0.0}^{1.0} 3^{x}\, dx=2.0/log(3) $$ (click integral calculator for calculations)

So, the error involved in the approximation of the integral is given as:

$$ {\left| \varepsilon \right| = \left| {\frac{{2 – 1.82}}{{2}}} \right| }\approx{ 0.08932 }={ 8.932\%} $$

You can subject to free online Simpson’s rule error calculator to verify the results.

Sometimes, it’s difficult to understand how to evaluate the area under the parabola. To deal with problems under such conditions, using a free simpson rule calculator is a reliable choice to make.

Let us see what we need to do:

**Input:**

- Write dow your function in a proper manner in the menu bar
- Make a selection of the variable w.r.t which you are going to evaluate integral
- Set lower and upper bound limits
- Select number of rectangles (must not an odd number)
- Click ‘calculate’

**Output:**

The Simpson calculator determines:

- Definite integral with the help of Simpson’s rule formula
- Evaluates the actual integral
- Calculates the error involved in the approximation process.

The major drawback of using Simpson’s rule is that if we have a function that is highly oscillatory or lacks derivatives at certain points, then this method is not suitable for finding accurate results. But making use of Simpson’s rule approximation calculator can make it possible to determine such integrals also.

Simpsons 1 / 8 and 3 / 8 are two cases of Newton’s cotes formulas. Simpsons 3/ 8rule requires the need for one more integral inside the integration range and gives lower error bounds.

The reason is that we use parabolas to approximate each part of the curve which is most efficient method in numerical analysis. Moreover, a free online simpson’s rule calculator provide you with the ease to determine the definite integral instantly along with each step involved in the calculations.

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.

Simpson’s rule is introduced only to determine the equation of the parabola most precisely. There is a vast use of Simpson’s rule calculator in the field of engineering and science as it provides a more absolute approximation of overall variation than either sum does alone. However, a Simpsons calculator also enables you to calculate the error of approximation.

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

From the source of lumen learning: Basic Integration Principles, Properties, Integration By Parts, Trigonometric Integrals