## Introduction to Trapezoidal Rule Calculator:

Trapezoidal rule calculator is an efficient tool that helps you to find the approximate value of the area under the curve in a bounded region. It is a beneficial tool for students, teachers, and professionals for making assignments, notes, or research reports.

Our trapezoid rule calculator is used to evaluate the definite integral function that divides into trapezoidal subintervals. It gives you solution of the given function in a few seconds without making any mistakes in evaluation.

## What is the Trapezoidal Rule?

Trapezoidal rule is a numerical integration process that is used to find the **estimated value** of the definite integral function in a bounded region. This method is specifically used to find the area under a graph by dividing it into subintervals.

Trapezoidal rule is the extension of the Reimann sum because it provides a more accurate solution of your given function without error using the Reimann sum rule.

## Formula of Trapezoidal Rule:

The **formula** of a trapezoidal rule is a continuous function f(x) over an interval [a, b] that divides the area into n smaller sections such as:

$$ \int_a^b f(x) dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] $$

- Δx: the change between the limit as Δx=b-a
- f(x): the definite function
- a and b: the upper and the lower limits

## How to Calculate the Trapezoidal Rule?

Trapezoidal rule is used to **calculate the definite integral** function in a closed region with the help of the trapezoidal rule formula to avoid complex calculation as the regular integral is divided by a rectangle instead of a subinterval.

**Step 1:** Determine the given function f(x), subinterval value, and the upper and the lower limit.

**Step 2:** Evaluate the interval with the help of the upper and the lower limit and n subinterval as Δx = b-a/n

**Step 3:** After finding all the values, add these values into the trapezoidal rule formula.

**Step 4:** Solve the function after adding values into a simplified form so that you get the solution of the trapezoidal rule function.

With our trapezoidal rule calculator, automate the calculation process of definite integrals in closed regions using a straightforward formula, and simplify complex calculations by approximating the integral with trapezoids instead of subintervals.

## Solved Example of Trapezoidal Rule:

The solved **example** of the trapezoidal rule will give you a better understanding of the trapezoidal rule and its evaluation procedure.

### Example: Find the first four subintervals:

$$ \int_0^1 \sqrt{1 + x^2} dx $$

**Solution**:

Determine the given data from the given function

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

Find the change in function Δx = b-a/n,

$$ \Delta x \;=\; \frac{b - a}{n}, \Delta x \;=\; \frac{1 - 0}{4} $$

$$ \Delta x \;=\; 0.25 $$

Find the given integral function subinterval n as,

$$ f(x) \;=\; f(x0),\; f(x_1),\; f(x_2),\; f(x_3),\; f(x_4) $$

$$ f(x) \;=\; f(x0),\; f(x_1),\; f(x_2),\; f(x_3),\; f(x_4) $$

$$ x_0 \;=\; 0, x_1 \;=\; 0.25, x_2 \;=\; 0.5, x_3 \;=\; 0.75, x_4 \;=\; 1 $$

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

$$ f(0.25) \;=\; \sqrt{1 + (0.25)^2} \approx 1.06066 $$

$$ f(0.5) \;=\; \sqrt{1 + (0.5)^2} \approx 1.11803 $$

$$ f(0.75) \;=\; \sqrt{1 + (0.75)^2} \approx 1.25708 $$

$$ f(1) \;=\; \sqrt{1 + 1^2} \;=\; \sqrt{2} \approx 1.41421 $$

Apply all the function values, change the limit in the trapezoidal rule formula,

$$ \int_0^1 \sqrt{1 + x^2} dx \approx \frac{0.25}{2} [1 + 2(1.06066 + 1.11803 + 1.25708 + 1.41421] $$

$$ \int_0^1 \sqrt{1 + x^2} dx \approx \frac{0.25}{2} [1 + 2(3.43577) + 1.41421] $$

$$ \int_0^1 \sqrt{1 + x^2} dx \approx \frac{0.25}{2} [1+ 6.87154 + 1.41421] $$

$$ \int_0^1 \sqrt{1 + x^2} dx \approx \frac{0.25}{2} \times 9.28575 $$

$$ \int_0^1 \sqrt{1 + x^2} dx \approx 1.16071875 $$

Therefore, the approximate value of the integral ∫01^{√1}+ x^{2}dx using the Trapezoidal Rule with n = 4 subintervals is 1.16071875.

For more convenience and accuracy in calculations, try our trapezoidal method calculator.

## How to Use the Trapezium Rule Calculator?

Trapezoidal calculator has an easy-to-use interface, so you can easily use it to evaluate the given definite integral problem. Before adding the input for the solutions of given trapezoidal rule problems, you must follow some simple steps. These steps are:

**Enter the definite function**in the input field that you want to evaluate using the trapezoidal rule.- Add the value of subinterval n for the trapezoidal rule problem in the input field.
- Add the value of upper and the lower limit for trapezoidal integral function.
- Recheck your input value for the trapezoidal rule problem before hitting the calculate button of trapezoidal approximation calculator to start the calculation process.
- Click on the “Calculate” button to get the desired result of your given Trapezoidal rule problem.
- If you want to try out our trapezoidal sum calculator to check its accuracy in solution, use the load example option.
- Click on the “Recalculate” button to get a new page for solving more trapezoidal rule questions.

## Final Result of Trapezoidal Method Calculator:

Trapezoid approximation calculator gives you the **solution** to a given definite integral problem when you give it an input value. It provides you with solutions that may contain as:

**Result Option**:

You can click on the result option, it provides you with a solution of trapezoidal rule questions.

**Possible Step**:

When you click on the possible steps option it provides you with the solution of the trapezoidal rule problem where steps are included.

## Benefits of Using Trapezoidal Rule Calculator:

Trapezoid rule calculator gives you multiple benefits whenever you use it to calculate trapezoidal rule problems to get its solution. These benefits are:

- Our trapezium rule calculator
**saves the time**and effort that you consume in solving trapezoidal rule questions. - It is a free-of-cost tool that helps you find the area of a graph using the trapezoidal rule.
- The trapezoidal calculator ensures highly accurate results when computing integrals, minimizing potential errors that can occur with manual methods.
- It is an adaptive tool that allows you to find the definite integral function.
- You can use this trapezoidal approximation calculator to practice different concepts easily.
- Trapezoidal sum calculator is a trustworthy tool that provides you with correct solutions as per your input to calculate the trapezoidal integral problem.