## Introduction to Simpson's Rule Calculator:

Simpson's rule calculator is an online tool that helps you to evaluate definite integral function's approx value. It is used to **find the area** under a curve from the given complex quadratic polynomial function in an interval.

When you solve the Simpson rule question manually it takes time or you may get stuck during calculations. That’s why we introduce simpson rule calculator that keeps you away from all these difficulties and provides you the solution with just one click.

## What is Simpson's Rule?

Simpson rule is a numerical integration process of calculating the approximate value of area under a graph over a bonded region. Simpson rule is the extension of the trapezoidal rule as it finds the complex integral problem of the second order (quadratic polynomial) functions.

Simpson rule has two formulas that are used to estimate the approximate value for different types of quadratic polynomial functions under a parabola.

## What is Simpson's 1/3 Rule?

For a function f(x) that divides intervals [a, b] into even number of subintervals, the Simpson's rule calculator uses **Simpson's 1/3 rule** formula as:

$$ \int_a^b f(x) dx \approx \frac{\Delta x}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)) $$

- h = b-a/n: the width of the subinterval/
- f(x): the function
- x0, x1, x2, x3…: the value of subinterval of a given function
- [a,b]: the interval of a given function

## What is Simpson's 3/8 Rule?

Simpson's 3/8 Rule is a method for approximating the cubic polynomials and **definite integral** of a function. It is particularly useful when the subintervals are a multiple of 3.

Simpson's 3/8 rule gives more accurate result than simpson's 1/3 rule but it is used for odd number values only. The simpsons rule calculator uses 3/8 rule also if the question is supposed to solve by the given formula,

$$ \int_a^{b} f(x) dx \;=\; \frac{3 \Delta x}{8} [f(x_0) + 2(f(x_3) + f(x_6) + f(x_{n-1})) + f(x_n)] $$

## How to Calculate Simpson's Rule?

Simpson's rule formulas is used to solve **complex integral** functions, quadratic polynomial functions or cubic polynomial functions. simpson 1/3 rule and simpson 3/8 rule have two separate formulas for complex calculations.

Let's see examples of the simpson rule where we would use both the rules solve a problem to understand how the simpson's 1/3 rule calculator solves such questions.

## Example of Simpson’s Rule:

An **example of simpson’s rule** is given below so that you may know how the simpson’s rule calculator works.

### Example: Find the following integral using simpson’s rule with n = 4.

$$ \int_0^4 (1 + x^2) dx $$

**Solution**:

**Step 1**:

Determine the values from the given integral function. In this example, the values are:

$$ n \;=\; 4,\; a \;=\; 0\; and\; b \;=\; 4,\; f(x) \;=\; 1 + x^2 $$

**Step 2**:

Find the width of subinterval h as h = b-a/n,

$$ h \;=\; \frac{4-0}{4} \;=\; 1 $$

$$ h \;=\; 1 $$

**Step 3**:

Then, find the subinterval of a given function f(x) such as:

$$ f(x) \;=\; f(x_0), f(x_1), f(x_2), f(x_3), f(x_4) $$

$$ x_0, x_1 \;=\; 1, x_2 \;=\; 2, x_3 \;=\; 3, x_4 \;=\; 4 $$

$$ f(0) \;=\; 1 + 0^2 \;=\; 1 $$

$$ f(1) \;=\; 1 + 1^2 \;=\; 2 $$

$$ f(2) \;=\; 1 + 2^2 \;=\; 5 $$

$$ f(3) \;=\; 1 + 3^2 \;=\; 10 $$

$$ f(4) \;=\; 1 + 4^2 \;=\; 17 $$

**Step 4**:

Apply the 1/3 Simpson rule and modify it as per the given function and its subintervals such as,

$$ \int_a^b f(x) dx \;=\; \frac{h}{3} [(y_0 + y_n) + 4(y_1 + y_3 + y_5 + … + y_{n-1}) + 2(y_2 + y_4 + y_6 + … + y_{n-2})] $$

**Step 5**:

Add all the values in the 1/3 Simpson rule formula to get the solution,

$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} [1 + 4(2 + 10) + 2(5) + 17] $$

**Step 6**:

Therefore, the solution of definite integral over a subinterval n = 4 is:

$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} [1 + 48 + 10 + 17] $$

$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} \times 76 \;=\; \frac{76}{3} \approx 25.333 $$

### Example: Find the following integral using Simpson’s ⅜ rule with n = 3.

$$ \int_0^3 e^x dx $$

**Solution**:

**Step 1**:

Determine the values from the given integral function. In this example, the values are,

$$ n \;=\; 3,\; a \;=\; 0\; and\; b \;=\; 3,\; f(x) \;=\; e^x $$

**Step 2**:

Find the width of subinterval h as:

$$ h \;=\; \frac{b-a}{n} $$

$$ h \;=\; \frac{3-0}{3} \;=\; 1 $$

$$ h \;=\; 1 $$

**Step 3**:

Then, find the subinterval of a given function f(x) such as:

$$ f(x) \;=\; f(x_0),\; f(x_1),\; f(x_2),\; f(x_3) $$

$$ x_0 \;=\; 0,\; x_1 \;=\; 1,\; x_2 \;=\; 2,\; x_3 \;=\; 3 $$

$$ f(0) \;=\; e^0 \;=\; 1 $$

$$ f(1) \;=\; e^1 \;=\; e $$

$$ f(2) \;=\; e^2 \;=\; e^2 $$

$$ f(3) \;=\; e^3 \;=\; e^3 $$

**Step 4**:

Apply the ⅜ Simpson rule and modify it and its subintervals as per the given function:

$$ \int_0^3 e^x dx \approx \frac{3h}{8} [f(0) + 3(f(1) + f(2)) + f(3)] $$

**Step 5**:

Add all the values in the 3/8 Simpson rule formula.

$$ \int_0^3 e^x dx \approx \frac{3.1}{8} [1 + 3(e + e^2) + e^3 ] $$

**Step 6**:

Therefore, the solution of definite integral over a subinterval n = 3 is,

$$ \int_0^3 e^x\; dx \approx \frac{3}{8} [1 + 3e + 3e^2 + e^3] $$

## How to Use Simpson's Rule Calculator?

The simpson rule calculator has a simple design that makes it easy for you to use it for evaluating the complex integral functions. You need to follow our guidelines before using it. These guidelines are:

**Enter the complex integral**function of the subinterval in the given input field.- Enter the value of the upper and lower limit of integral of simpson rule in the input field.
- Enter the subinterval value in the required field.
- Recheck the given integral expression before clicking the calculate button of simpsons rule calculator.
- Click the “Calculate” button to get the result of your given complex integral over a single interval.
- If you are trying our simpson's 1/3 rule calculator for the first time then you can use the load example option to learn more about this concept.
- Click on the “Recalculate” button to get a new page for finding solutions of complex integral problems using the Simpson rule.

## Final Result of Simpson Rule Calculator:

When you add the input, the simpson calculator gives you the solution from the given integral function. The results are shown as:

**Result Option**:

When you click on the result option the simpson 1/3 rule calculator gives you the **solution** of the integral function.

**Possible Steps**:

When you click on it, this option will give you step by step solution of simpson's rule problem.

## Advantages of Simpsons Rule Calculator:

The simpson's rule error calculator provides you with many advantages and helps you to calculate integral problems. These advantages are:

- The simpson's method calculator is a
**free tool**so you can use it to find the solution of integral for free. - It is a manageable tool that can manage various types of integral problems that have quadratic polynomial or cubic functions. It conatins a built-in Simpson rule to solve these problems easily.
- Our simpson calculator gives you conceptual clarity for the integral process by giving you an oppportunity to solve multiple examples.
- It saves the time that you consume in the calculation of complex integral problems and provides solutions in a few seconds.
- Simpson 1/3 rule calculator is a reliable tool that provides you accurate solutions of integral problems without any errors.
- You can use the simpson's rule calculator multiple times for evaluating antiderivative problems over a bounded region.