Boole's Rule of Numerical Integration

Boole's Rule is a 4th-degree Newton-Cotes closed integration formula that approximates the definite integral $\int_{a}^{b} f(x) \, dx$ by fitting a 4th-order polynomial across 5 equally spaced points over every subinterval block.

A fundamental requirement of Boole's Rule is that the number of subintervals $n$ must be a multiple of 4 (e.g. $n = 4, 8, 12$ ).

Formula for Boole's Rule:

I \approx \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 + \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 4).
$h = \frac{b - a}{n}$ is the step size.

Example of Boole's Rule

Evaluate Definite Integral

\int_{0}^{4} e^x , dx

With $n = 4 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

h = \frac{4 - 0}{4} = 1.0

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4

Step 3: Evaluate Function Values ( $y_i = e^{x_i}$ )

y_0 = 1.0, \quad y_1 = 2.71828, \quad y_2 = 7.38906

y_3 = 20.08554, \quad y_4 = 54.59815

Step 4 & 5: Apply Boole's Formula

I \approx \frac{2(1.0)}{45} \left[ 7(1) + 32(2.71828) + 12(7.38906) + 32(20.08554) + 7(54.59815) \right]

I \approx \frac{2}{45} \left[ 1207.8767 \right] \approx 53.6834

Conclusion

\boxed{\int_{0}^{4} e^x , dx \approx 53.6834}

Therefore, using Boole's Rule with $n = 4$ , the approximate value of the integral is 53.6834 (exact value $e^4 - 1 \approx 53.59815$ ).

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Boole's Rule of Numerical Integration

A fundamental requirement of Boole's Rule is that the number of subintervals $n$ must be a multiple of 4 (e.g. $n = 4, 8, 12$ ).

Formula for Boole's Rule:

I \approx \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 + \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 4).
$h = \frac{b - a}{n}$ is the step size.

Example of Boole's Rule

Evaluate Definite Integral

\int_{0}^{4} e^x , dx

With $n = 4 \text{ subintervals}$

Step 1: Find Step Size ( $h$ )

h = \frac{4 - 0}{4} = 1.0

Step 2: Determine Grid Points ( $x_i$ )

x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4

Step 3: Evaluate Function Values ( $y_i = e^{x_i}$ )

y_0 = 1.0, \quad y_1 = 2.71828, \quad y_2 = 7.38906

y_3 = 20.08554, \quad y_4 = 54.59815

Step 4 & 5: Apply Boole's Formula

I \approx \frac{2(1.0)}{45} \left[ 7(1) + 32(2.71828) + 12(7.38906) + 32(20.08554) + 7(54.59815) \right]

I \approx \frac{2}{45} \left[ 1207.8767 \right] \approx 53.6834

Conclusion

\boxed{\int_{0}^{4} e^x , dx \approx 53.6834}

Therefore, using Boole's Rule with $n = 4$ , the approximate value of the integral is 53.6834 (exact value $e^4 - 1 \approx 53.59815$ ).

Interactive Boole's Rule Calculator

Boole's Rule Solver

PreviousSimpson 3/8 Rule

Unit 3•15 of 28

NextWeddle's Rule

Boole's Rule of Numerical Integration

Example of Boole's Rule

Step 1: Find Step Size (hhh)

Step 2: Determine Grid Points (xix_ixi​)

Step 3: Evaluate Function Values (yi=exiy_i = e^{x_i}yi​=exi​)

Step 4 & 5: Apply Boole's Formula

Conclusion

Interactive Boole's Rule Calculator

Boole's Rule Solver

NETZ Engine

Boole's Rule of Numerical Integration

Example of Boole's Rule

Step 1: Find Step Size (hhh)

Step 2: Determine Grid Points (xix_ixi​)

Step 3: Evaluate Function Values (yi=exiy_i = e^{x_i}yi​=exi​)

Step 4 & 5: Apply Boole's Formula

Conclusion

Interactive Boole's Rule Calculator

Boole's Rule Solver

Step 1: Find Step Size ( $h$ )

Step 2: Determine Grid Points ( $x_i$ )

Step 3: Evaluate Function Values ( $y_i = e^{x_i}$ )

Step 1: Find Step Size ( $h$ )

Step 2: Determine Grid Points ( $x_i$ )

Step 3: Evaluate Function Values ( $y_i = e^{x_i}$ )