# Simpson's Rule Calculator

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##### by Alan Walker

Last Updated December 24, 2022

## Introduction to Simpson's Rule Calculator

Simpson's integration calculator is an online tool that evaluates definite integrals. It follows the numerical integration to approximate the value of a definite integral by using quadratic equations. It collects the user's input function and provides quick results.

In the numerical method, Simpson's rule is a technique for finding antiderivatives using quadratic equations. We introduce you to simpson 1/3 rule calculator that follows Simpson's rule formula and makes calculations easy. So, you can find the definite integral approximation with just a few clicks.

## The formula used by Simpson's 3/8 Rule Calculator

Simpson's rule is a technique for approximating integrals in numerical integration. Using this technique, a function is integrated between two points, a and b. It is based on the fundamental theorem of calculus and used to approximate such integrals that usual techniques cannot integrate.

The Simpson's rule volume calculator uses the given function and the interval to approximate it. Using this tool, you can calculate the area under a parabola or a curve. It provides accurate area by using Simpson's 1/3 and 3/8 formulas.

The Simpson’s rule formula is:

$$\int^b_af(x)dx=\frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+\cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$$

Where,

• $$h=\frac{b-a}{n}$$
• $$x_0=a \quad,\quad x_n=b$$

This formula is known as Simpson's 1/3 rule due to the factor 1/3. It is applicable for quadratic polynomials. This formula can also be written as:

$$\int^b_af(x)dx=\frac{h}{3}\left[(y_0+y_1)+4(y_1+y_3+\cdots)+2(y_2+y_4+\cdots)\right]$$

Simpson's 3/8 rule formula is similar to Simpson's 1/3. There is only one difference between 1/3 and 3/8. It is:

$$\int^b_af(x)dx=\frac{3h}{8}\left[(y_0+y_n)+3(y_1+y_2+y_4+\cdots)+2(y_3+y_6+\cdots)\right]$$

It approximates the function by using 3rd order polynomial. Simpson's rule approximation calculator uses the above formulas.

The approximation of an integral can also be calculated by using midpoint rule. It approximates the integral by midpoints.

## How to use Simpson's Integration Calculator?

Making calculations with simpson's rule integral calculator is very easy. You have to follow some simple steps. These steps will help you to find the area bounded by two curves in a step-by-step way.

1. Enter the function for which you want to approximate.
2. Since simpson's rule calculator with points approximates definite integrals, so for this you have to provide the upper and lower bound value.
3. Enter the width of the interval.
4. Now choose the variable of integration, i.e., x, y, or z.
5. Review the input value and click the calculate button.

After clicking the calculate button, the Simpson's rule calculator will provide your approximation of the given function within a few seconds. Also calculate approximated sum of a function by using right riemann sum calculator.

## Why use Simpson's 1/3 Rule Calculator?

In calculus, many different formulas are used to approximate curves by using numerical integration. It involves Simpson's 1/3, Simpson’s 3/8 and trapezoidal rule. But these methods are similar to each other. Therefore, it is not simple to use these rules manually. Our Simpson's rule error calculator can assist you in doing calculations efficiently.

While doing calculations by hand, you may need clarification on the two rules, i.e., Simpson's 1/3 and Simpson's 3/8 rule. Also, you can get wrong calculations due to using the wrong formula. Therefore, it would be best for you to use Simpson's rule Calculator.

## Benefits of using Error Bound Calculator for Simpson's Rule

The use of Simpson's rule integral calculator has various benefits. Most complex functions that are impossible to approximate by standard integrals can be solved or calculated using this calculator. Some of the benefits of this calculator are:

• This calculator will solve any function using Simpson's rule in just a few seconds.
• It can save you time and energy that you use in manual calculations.
• It can handle both quadratic and cubic polynomials as well.
• Composite simpson's rule calculator is free to use, and you don't have to pay for other premium tools to learn Simpson's rule.

We hope that you find our tool helpful for you. We also offer you many other integral tools that can help you to learn integration easily. Also you can learn and do calculations for volume of revolution by using our volume by disk method calculator.

## FAQ's

### What is meant by Simpson’s rule?

Simpson’s rule is a technique of approximating integrals in numerical integration. By using this technique a function is integrated between two points a and b. The Simpson’s rule formula is written as:

$$\int^b_af(x)dx=\frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+\cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$$

### Why is Simpson's Rule important?

Since Simpson’s rule follows the fundamental concept of integration to solve complex integrals. This rule is important to find approximation of curves that involves numerical integration. It is because this rule can approximate the integrals up to 3rd order.

### Is Simpson's rule a Riemann sum?

A technique for estimating definite integrals of functions is Simpson's rule. It is exact for linear and quadratic functions and typically (but not always) more accurate than approximations made using Riemann sums or the trapezium rule.