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# Trapezoidal Rule Calculator

To calculate the area under the curve, enter the function, upper & lower bounds, and the subintervals of the entered function in the trapezoidal rule calculator.

Table of Contents:

## Trapezoidal rule calculator

Trapezoidal rule calculator is used to calculate the approximated integral with the help of the lower & upper bound and the interval of the function.

## What is the trapezoidal rule?

The trapezoidal rule is a numerical integration method used to approximate the value of a definite integral by dividing the area under the curve into trapezoids and summing up the areas of those trapezoids.

## Formula of Trapezoidal rule:

The formula for the trapezoidal rule is as follows:

_{a}∫^{b} f(x) dx ≈ [Δx / 2] [f(x_{0}) + 2{f(x_{1}) + f(x_{2}) + ... + f(x_{n-1})) + f(x_{n})}]

In the above formula:

- Δx = (b – a) / n
- n is the number of trapezoids, used to approximate the area under the curve.

## How to calculate the problems of trapezoidal rule?

**Example 1: **

Calculate the area under the curve _{0}∫^{1} e^{-x^}^{2} dx; if n = 4.

**Solution:**

**Step 1: **Calculate “Δx”

Using the trapezoidal rule with n = 4, we have:

Δx = (b – a) / n

Δx = (1 – 0) / 4

Δx = 0.25

**Step 2:** Calculate the values of the function

f (0) = e^{0}

f (0) = 1

f (0.25) = e^{ (-0.25)}^{2}

f (0.25) ≈ 0.968

f (0.5) = e^{ (-0.5)}^{2}

f (0.5) ≈ 0.882

f (0.75) = e^{ (-0.75)}^{2}

f (0.75) ≈ 0.705

f (1) = e^{ (-1)}^{2}

f (1) ≈ 0.368

**Step 3:** Put the values in the trapezoidal rule formula.

Using the trapezoidal rule formula, we have:

_{0}∫^{1} e^{-x}^{2} dx ≈ [(1 – 0) / (2*4)] [1 + 2(0.968) + 2(0.882) + 2(0.705) + 0.368]

_{0}∫^{1} e^{-x}^{2} dx ≈ 0.7468

Therefore, the approximate value of the integral using the trapezoidal rule with n = 4 is **0.7468**.