Trapezoidal Rule of Numerical Integration

The Trapezoidal Rule is a fundamental numerical integration technique used to find an approximate value of a definite integral $\int_{a}^{b} f(x) \, dx$ by approximating the region under the graph of the function $f(x)$ as a series of trapezoids.

It is particularly useful when the function $f(x)$ is difficult or impossible to integrate analytically, or when data points are obtained experimentally at discrete intervals.

Formula for Trapezoidal Rule:

I = \int_{a}^{b} f(x) , dx \approx \frac{h}{2} \left[ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \right]

Where:

$a$ is the lower limit of integration.
$b$ is the upper limit of integration.
$n$ is the number of subintervals.
$h = \frac{b - a}{n}$ is the step size or width of each subinterval.
$y_0, y_1, \\dots, y_n$ are the function values evaluated at grid points.

Example of Trapezoidal Rule

Evaluate Definite Integral

\int_{0}^{1} \frac{1}{1 + x^2} , dx

With $n = 6 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

Given $a = 0, b = 1, n = 6$ :

h = \frac{b - a}{n} = \frac{1 - 0}{6} = \frac{1}{6} \approx 0.16667

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0.0, \quad x_1 = 0.1667, \quad x_2 = 0.3333

x_3 = 0.5, \quad x_4 = 0.6667, \quad x_5 = 0.8333, \quad x_6 = 1.0

Step 3: Evaluate Function Values ( $y_i = \frac{1}{1 + x_i^2}$ )

y_0 = 1.00000, \quad y_1 = 0.97297

y_2 = 0.90000, \quad y_3 = 0.80000

y_4 = 0.69231, \quad y_5 = 0.59016, \quad y_6 = 0.50000

Step 4 & 5: Apply Formula

I \approx \frac{1/6}{2} \left[ (1.0 + 0.5) + 2(3.95544) \right]

I \approx \frac{1}{12} \left[ 9.41088 \right] \approx 0.78424

Conclusion

\boxed{\int_{0}^{1} \frac{1}{1 + x^2} , dx \approx 0.78424}

Therefore, using Trapezoidal Rule with $n = 6$ , the approximate value of the integral is 0.78424 (exact value $\frac{\pi}{4} \approx 0.78540$ ).

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Trapezoidal Rule of Numerical Integration

It is particularly useful when the function $f(x)$ is difficult or impossible to integrate analytically, or when data points are obtained experimentally at discrete intervals.

Formula for Trapezoidal Rule:

I = \int_{a}^{b} f(x) , dx \approx \frac{h}{2} \left[ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \right]

Where:

$a$ is the lower limit of integration.
$b$ is the upper limit of integration.
$n$ is the number of subintervals.
$h = \frac{b - a}{n}$ is the step size or width of each subinterval.
$y_0, y_1, \\dots, y_n$ are the function values evaluated at grid points.

Example of Trapezoidal Rule

Evaluate Definite Integral

\int_{0}^{1} \frac{1}{1 + x^2} , dx

With $n = 6 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

Given $a = 0, b = 1, n = 6$ :

h = \frac{b - a}{n} = \frac{1 - 0}{6} = \frac{1}{6} \approx 0.16667

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0.0, \quad x_1 = 0.1667, \quad x_2 = 0.3333

x_3 = 0.5, \quad x_4 = 0.6667, \quad x_5 = 0.8333, \quad x_6 = 1.0

Step 3: Evaluate Function Values ( $y_i = \frac{1}{1 + x_i^2}$ )

y_0 = 1.00000, \quad y_1 = 0.97297

y_2 = 0.90000, \quad y_3 = 0.80000

y_4 = 0.69231, \quad y_5 = 0.59016, \quad y_6 = 0.50000

Step 4 & 5: Apply Formula

I \approx \frac{1/6}{2} \left[ (1.0 + 0.5) + 2(3.95544) \right]

I \approx \frac{1}{12} \left[ 9.41088 \right] \approx 0.78424

Conclusion

\boxed{\int_{0}^{1} \frac{1}{1 + x^2} , dx \approx 0.78424}

Therefore, using Trapezoidal Rule with $n = 6$ , the approximate value of the integral is 0.78424 (exact value $\frac{\pi}{4} \approx 0.78540$ ).

Interactive Trapezoidal Rule Calculator

Trapezoidal Rule Solver

PreviousNewton Divided Difference

Unit 3•12 of 28

NextSimpson 1/3 Rule

Trapezoidal Rule of Numerical Integration

Example of Trapezoidal Rule

Step 1: Find Step Size (hhh)

Step 2: Determine Grid Points (xix_ixi​)

Step 3: Evaluate Function Values (yi=11+xi2y_i = \frac{1}{1 + x_i^2}yi​=1+xi2​1​)

Step 4 & 5: Apply Formula

Conclusion

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Trapezoidal Rule Solver

NETZ Engine

Trapezoidal Rule of Numerical Integration

Example of Trapezoidal Rule

Step 1: Find Step Size (hhh)

Step 2: Determine Grid Points (xix_ixi​)

Step 3: Evaluate Function Values (yi=11+xi2y_i = \frac{1}{1 + x_i^2}yi​=1+xi2​1​)

Step 4 & 5: Apply Formula

Conclusion

Interactive Trapezoidal Rule Calculator

Trapezoidal Rule Solver

Step 1: Find Step Size ( $h$ )

Step 2: Determine Grid Points ( $x_i$ )

Step 3: Evaluate Function Values ( $y_i = \frac{1}{1 + x_i^2}$ )

Step 1: Find Step Size ( $h$ )

Step 2: Determine Grid Points ( $x_i$ )

Step 3: Evaluate Function Values ( $y_i = \frac{1}{1 + x_i^2}$ )