## Introduction to Midpoint Rule Calculator:

Midpoint rule calculator is an online tool that helps you to **evaluate the approx value** of a given area over a closed interval using midpoint rule. It is used to compute the estimated value of each subinterval of a given area in a graph.

Our midpoint rule online calculator is a useful online source to find the definite integral of each subinterval. It especially solve the definite integral problems which cannot be solved easily by analytic calculus methods.

## What is the Midpoint Rule?

Midpoint rule is a numeral analysis method in which you can calculate the **estimated value** of an area over a bounded region when you divide it with each of its subintervals. It is known as midpoint rule or midpoint approximate.

It is a type of riemann sum integral method in which you find the area of a rectangle from the given interval.

## Formula of Midpoint Rule:

The **formula of midpoint rule** consists of a continuous function f(x) over a closed interval [a, b] that divides each subinterval.

$$ \int_a^b f(x) dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*) $$

- a and b: the lower or the upper limits
- Δx: change in each subinterval
- f(x): the continuous function of definite integral
- n: the number of subinterval its values vary as n = 0,1,2,....

## How to Calculate Midpoint Rule For Integrals?

To **calculate the midpoint rule** for integrals in steps, you shoudl use the midpoint rule formula. Here is a step-by-step guide about how to solve the given midpoint question.

**Step 1**: Identify the function f(x) that you want to integrate and the interval [a, b] over which you want to integrate.

**Step 2**: Choose the number of subintervals n that divide the interval [a, b].

**Step 3**: Calculate the width of each subinterval Δx,

$$ \Delta x \;=\; \frac{b-a}{n} $$

**Step 4**: For each subinterval i, the midpoint is calculated as:

$$ x_i^* \;=\; x_0 + x_{\frac{1}{2}}, x_1 + x_{\frac{2}{2}}, x_2 + x_{\frac{3}{2}},... $$

$$ For\; i \;=\; 1, 2,...,n $$

**Step 5**: Put the above values in the midpoint rule formula and simplify it to get a solution.

$$ M_n \;=\; \sum_{i=1}^{n} f(m_i) \Delta x $$

**Step 6**:Evaluate the given integral and apply its upper and lower limits in solution.

**Step 7**: The final approximation of the integral is given by,

$$ Error\; in\; M_n ≤ \frac{M(b-a)^3}{24n^2} $$

Alternatively, You can use our midpoint rule calculator, which is highly efficient at finding the definite integral for each subinterval. It provides accurate results within seconds.

## Practical Example of Midpoint Rule:

An **example** of the midpoint rule with a solution is given below to understand how the midpoint rule integral calculator works.

### Example: Use Midpoint Rule to Estimate the Following Using Four Subintervals

$$ \int_0^1 x^2 dx $$

**Solution**:

Identify the given function f(x) and limits,

$$ f(x) \;=\; x^2,\; a \;=\; 0,\; b \;=\; 1 $$

$$ \int_0^1 x^2 dx $$

Find the value of each subinterval,

$$ Δx \;=\; \frac{b-a}{n} $$

$$ \Delta x \;=\; \frac{1-0}{4} \;=\; \frac{1}{4} $$

As the limit is between 0 and 1 so the subinterval is between (0,1),

$$ [0, \frac{1}{4}],\; [\frac{1}{4}, \frac{1}{2}], [\frac{1}{2}, \frac{3}{4}],\; and\; [\frac{3}{4}, 1] $$

For f(x*) value we use this rule f(x0 + x1/2) for each subinterval,

$$ \int_a^b f(x) \;=\; \Delta x \left(f \left( \frac{x_0 + x_1}{2} \right) + f \left(\frac{x_1 + x_2}{2} \right) + f \left(\frac{x_2 + x_3}{2} \right) + … + f \left(\frac{x_{n-2} + x_{n-1}}{2} \right) + f \left(\frac{x_{n-1} + x_{n}}{2} \right) \right) $$

The midpoints of these subintervals are,

$$ \left( \frac{1}{8},\; \frac{3}{8},\; \frac{5}{8},\; \frac{7}{8} \right) $$

The formula of midpoint rule,

$$ \int_a^b f(x) dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*) $$

Put the value of Δx and f(x*) in the above formula,

$$ M_4 \;=\; \frac{1}{4} . f (\frac{1}{8} ) + \frac{1}{4} . f (\frac{3}{8}) + \frac{1}{4} . f (\frac{5}{8}) + \frac{1}{4} . f(\frac{7}{8}) $$

Simplify it,

$$ \frac{1}{4} . \frac{1}{64} + \frac{1}{4} . \frac{9}{64} + \frac{1}{4} . \frac{25}{64} + \frac{1}{4} . \frac{21}{64} \;=\; \frac{21}{64} $$

After evaluating the limit of the given function we get,

$$ \int_0^1 x^2 dx \;=\; \frac{1}{3} $$

For the error approximation value,

$$ \biggr| \frac{1}{3} - \frac{21}{64} \biggr| \;=\; \frac{1}{192} \approx 0.0052 $$

## How to Use the Midpoint Rule Approximation Calculator?

Midpoint rule integration calculator has a simple design that makes it easy for you for evaluating the midpoint rule problems. Follow some simple steps that are given as:

- Enter the given definite integral function in the given input field (that you want to evaluate to find its approx value).
- Add the upper and the lower limit of the midpoint integral in the next input field.
- Add the number of subinterval around which you evaluate the approx value.
- Check the given integral function value before clicking on the calculate button of midpoint rule formula calculator to start the evaluation process.
- Click the “
**Calculate**” button to get the result of your given midpoint rule problem. - If you are trying our midpoint rule error calculator for the first time then you can use the load example to learn more about this concept.
- The “Recalculate” button brings back to the home page where you can find more solutions of midpoint rule problems.

## Output from Midpoint Rule Online Calculator:

The midpoint rule calculator gives you the **solution** from a given function when you give it an input. It may included as:

**Result Option**:

When you click on the result option, it gives you a solution of definite integral function for each interval.

**Possible Steps**:

When you click on it, this option will provide you step by step solution of the estimated value of definite integral.

**Plot Option**:

It will give you a graphical representation of a definite integral function solution.

## Benefits of Using Midpoint Rule Integral Calculator:

Midpoint rule approximation calculator provides you with tons of benefits as it helps you to calculate the estimated value of each subinterval of a given function and give you solutions without any trouble. These benefits are:

- It is a free-of-cost tool so you can use it for free to find midpoint problem with solutions without spending.
- Midpoint rule integration calculator gives you conceptual clarity for the midpoint rule process when you use it for practicing more examples.
- It saves the time and effort that you consume on the calculation of complex definite integral functions for finding the estimated value manually.
- Midpoint rule formula calculator is an
**adaptable tool**that provides you with accurate solutions whenever you use it to calculate integral for each subinterval without any mistakes in the evaluation process. - Our tool enables you to use it multiple times for the evaluation of midpoint rule problems.
- Midpoint rule error calculator is a handy tool because you can access it through an online platform from anywhere with the help of the internet only.