## Introduction to Riemann Sum Calculator with Steps:

Midpoint Riemann sum calculator is an amazing online source that helps you to find the integral of n subintervals. It is used to **find the solution** of the sum of the area by dividing it into n subinterval of that area.

The riemann sum formula calculator is a helpful tool for everyone who want to get the solution of Riemann sum problems quickly and easily.

## What is Riemann Sum?

Riemann sum is a numerical integration method used to evaluate the area under the curve questions which is divided into n number of rectangles.

It uses the Riemann sum formula to find the solution of the given integral function means it helps in solving complex integral functions. Riemann sum gives a more accurate result of a definite integral function than the trapezoidal rule.

## Formula of Riemann Sum:

The **formula of Riemann sum** consists of a function f(x) in the interval [a, b] such as it divides it into n subinterval Δx. The formula used by the left riemann sum calculator is given below,

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

Whereas,

$$ i \;=\; 0,\; 1,\; 2,\; 3… $$

## How to Calculate Riemann Sum?

The upper and lower sum calculator **calculates the approximate value** of the area under a curve and divide n into smaller rectangles in calculus and numerical analysis. The Riemann sum calculation process is given as:

**Step 1**:

Identify the area under a curve f(x) over an interval [a, b] and subinterval value.

**Step 2**:

Determine in how many subintervals n you will divide the interval [a, b]. The more subintervals you use, the more accurate approximate value you get.

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

**Step 3**:

Choose which method is best for the given function: the left riemann sum, right riemann sum, or midpoint riemann sum rule.

**Left Riemann Sum**:

Sum the areas of rectangles using the left endpoint of each subinterval. The formula behind the riemann sum to integral calculator is,

$$ L_n \;=\; \sum_{i = 1}^{n} f(x_{i - 1}) \Delta x $$

**Right Riemann Sum**:

Sum the **areas of rectangles** using the right endpoint of each subinterval,

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

**Midpoint Riemann Sum**:

Sum the areas of rectangles using the midpoint of each subinterval. The midpoint riemann sum formula used by the riemann sum limit calculator is,

$$ M_n \;=\; sum_{i=1}^{n} f \biggr( \frac{x_{i-1} + x_i}{2} \biggr) \Delta x $$

**Step 3**:

Put the require values of f(x) as per your chosen formula of reimann sum.

## Solved Example of Riemann Sum:

The left riemann sum calculator can help you **solve different problems** but it is essential to solve such problems manually so an example is given below.

**Example**:

Approximate the following integral using the left Riemann sum with 4 subintervals.

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

**Solution**:

Given data is:

The function is, $$ f(x) \;=\; 2x + 1 $$

Interval is [0, 4], n = 4

**Calculate the width of each subinterval**:

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

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

$$ \Delta x \;=\; 1 $$

The left Reimann sum formula is,

$$ L_4 \;=\; \sum_{i=1}^{4} f(x_{i = 1}) \Delta x $$

Find xi and f(xi) value such as,

$$ x_0 \;=\; 0, x_1 \;=\; 1, x_2 \;=\; 2, x_3 \;=\; 3, x_4 \;=\; 4 $$

$$ f(x_0) \;=\; f(0) \;=\; 2 . 0 + 1 \;=\; 1 $$

$$ f(x_1) \;=\; f(1) \;=\; 2.1 + 1 \;=\; 3 $$

$$ f(x_2) \;=\; f(2) \;=\; 2.2 + 1 \;=\; 5 $$

$$ f(x_3) \;=\; f(3) \;=\; 2.3 + 1 \;=\; 7 $$

Add these values in the left Riemann sum formula,

$$ L_4 \;=\; f(x_0) \Delta x + f(x_1) \Delta x + f(x_2) \Delta x + f(x_3) \Delta x $$

Simplify the above expression to get the solution of given problem,

$$ L_4 \;=\; 1.1 + 3.1 + 5.1 + 7.1 $$

$$ L_4 \;=\; 1 + 3 + 5 + 7 $$

$$ L_4 \;=\; 16 $$

The approx value of given reimann sum value is 16.

## How to Use Midpoint Riemann Sum Calculator?

The riemann sum formula calculator has a simple design that enables you to easily calculate the definite integral over n interval questions. Before adding the input value for solutions, you must follow some instructions. These instructions are:

**Enter the definite integral**function in the input box for Riemann sum solution.- Enter the upper and the lower limits for reimann sum function in the input box.
- Recheck your input function value before hitting the calculate button of upper and lower sum calculator.
- Click on the “Calculate” button to get the desired result of your given Riemann Sum question.
- If you are trying our riemann sum to integral calculator for the first time then you can use the load example option.
- Click on the “Recalculate” button to get a new page for solving more riemann sum problems.

## Results from Riemann Sum Formula Calculator:

The right riemann sum calculator with steps gives you the **solution** of the integral problem when you add the input. It provides you solutions of the given function problem. It may contain as:

**Result Option**:

When you click on the result option, it provides you with a solution for the reimann sum integral function.

**Possible Step**:

When you click on the possible steps option, it provides you with the solution of Riemann sum problems in steps.

## Benefits of Upper and Lower Sum Calculator:

The riemann sum table calculator has many uses that you obtain whenever you use it to calculate definite integral problems for getting the approximate value of the integral function in solutions. These benefits are:

- The riemann sum limit calculator is an adaptable tool that can be operated through electronic devices (like laptops, computers, mobile, tablets, etc) with the help of the internet.
- Our tool saves the time and effort that you consume in doing complex calculations of the Riemann sum problems.
- The riemann sum formula calculator is a
**free tool**so you can use it to find the Riemann sum problems easily. - It provides you solution of Riemann sum questions with a complete process in a step-by-step method for a better understanding.
- It is an educational tool so you can use this right riemann sum calculator with steps for practicing various riemann sum problems.