Trapezoidal Rule Calculator

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(x₀) + 2{f(x₁) + f(x₂) + ... + 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) = e⁰

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^-x2 dx ≈ [(1 – 0) / (2*4)] [1 + 2(0.968) + 2(0.882) + 2(0.705) + 0.368]

₀∫¹ e^-x2 dx ≈ 0.7468

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