Weddle's Rule of Numerical Integration

Weddle's Rule is a highly accurate numerical integration method derived from Newton-Cotes formulas by using a 6th-order polynomial approximation across 7 equally spaced points over every block.

A key requirement of Weddle's Rule is that the total number of subintervals $n$ must be a multiple of 6 (e.g. $n = 6, 12, 18$ ).

Formula for Weddle's Rule:

I \approx \frac{3h}{10} \left[ y_0 + 5y_1 + y_2 + 6y_3 + y_4 + 5y_5 + 2y_6 + \dots \right]

Where:

$a$ is the lower limit of integration.
$b$ is the upper limit of integration.
$n$ is the number of subintervals (must be a multiple of 6).
$h = \frac{b - a}{n}$ is the step size.

Example of Weddle's Rule

Evaluate Definite Integral

\int_{0}^{6} \frac{1}{1 + x^2} , dx

With $n = 6 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

h = \frac{6 - 0}{6} = 1.0

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3

x_4 = 4, \quad x_5 = 5, \quad x_6 = 6

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

y_0 = 1.0000, \quad y_1 = 0.5000, \quad y_2 = 0.2000

y_3 = 0.1000, \quad y_4 = 0.0588, \quad y_5 = 0.0385, \quad y_6 = 0.0270

Step 4 & 5: Apply Weddle's Formula

I \approx \frac{3(1.0)}{10} \left[ 1.0 + 5(0.5) + 0.2 + 6(0.1) + 0.0588 + 5(0.0385) + 0.0270 \right]

I \approx 0.3 \times 4.5773 \approx 1.3732

Conclusion

\boxed{\int_{0}^{6} \frac{1}{1 + x^2} , dx \approx 1.3732}

Therefore, using Weddle's Rule with $n = 6$ , the approximate value of the integral is 1.3732 (exact value $\arctan(6) \approx 1.4056$ ).

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Weddle's Rule of Numerical Integration

Weddle's Rule is a highly accurate numerical integration method derived from Newton-Cotes formulas by using a 6th-order polynomial approximation across 7 equally spaced points over every block.

A key requirement of Weddle's Rule is that the total number of subintervals $n$ must be a multiple of 6 (e.g. $n = 6, 12, 18$ ).

Formula for Weddle's Rule:

I \approx \frac{3h}{10} \left[ y_0 + 5y_1 + y_2 + 6y_3 + y_4 + 5y_5 + 2y_6 + \dots \right]

Where:

$a$ is the lower limit of integration.
$b$ is the upper limit of integration.
$n$ is the number of subintervals (must be a multiple of 6).
$h = \frac{b - a}{n}$ is the step size.

Example of Weddle's Rule

Evaluate Definite Integral

\int_{0}^{6} \frac{1}{1 + x^2} , dx

With $n = 6 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

h = \frac{6 - 0}{6} = 1.0

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3

x_4 = 4, \quad x_5 = 5, \quad x_6 = 6

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

y_0 = 1.0000, \quad y_1 = 0.5000, \quad y_2 = 0.2000

y_3 = 0.1000, \quad y_4 = 0.0588, \quad y_5 = 0.0385, \quad y_6 = 0.0270

Step 4 & 5: Apply Weddle's Formula

I \approx \frac{3(1.0)}{10} \left[ 1.0 + 5(0.5) + 0.2 + 6(0.1) + 0.0588 + 5(0.0385) + 0.0270 \right]

I \approx 0.3 \times 4.5773 \approx 1.3732

Conclusion

\boxed{\int_{0}^{6} \frac{1}{1 + x^2} , dx \approx 1.3732}

Therefore, using Weddle's Rule with $n = 6$ , the approximate value of the integral is 1.3732 (exact value $\arctan(6) \approx 1.4056$ ).

Interactive Weddle's Rule Calculator

Weddle's Rule Solver

PreviousBoole's Rule

Unit 3•16 of 28

NextNumerical Differentiation

Weddle's Rule of Numerical Integration

Example of Weddle's 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 Weddle's Formula

Conclusion

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Weddle's Rule Solver

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Weddle's Rule of Numerical Integration

Example of Weddle's 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 Weddle's Formula

Conclusion

Interactive Weddle's Rule Calculator

Weddle's 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 ( $y_i = \frac{1}{1 + x_i^2}$ )