UNIT 3 • NUMERICAL INTEGRATION

Boole's Rule
Quartic Integration

Boole's Rule fits a 4th-order quartic polynomial across blocks of 5 points (4 subintervals), providing high-order numerical integration precision when total subintervals $n$ is a multiple of 4!

BOOLE'S CLOSED 4TH-ORDER FORMULAMULTIPLE OF 4 REQUIREMENT

I \approx \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 + \dots \right]

1. Pre-factor (2h / 45)2 times step size over 45:

h = \frac{b-a}{n}

2. Weight PatternBlock weights repeat in sequence: 7, 32, 12, 32, 7.

3. Shared BoundariesBlock boundaries sum:

7 + 7 = 14

at interior segment junctions.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the definite exponential integral:

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

Using $n = 4$ subintervals (multiple of 4)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 4$ and $n = 4$ :

h = \frac{b - a}{n} = \frac{4 - 0}{4} = 1.0

STEP 02 • TABULATE FUNCTION VALUES & BOOLE WEIGHTS

i	x_i	y_i = e^(x_i)	Boole Weight (w_i)
0	0.0	1.00000 (y_0)	7
1	1.0	2.71828 (y_1)	32
2	2.0	7.38906 (y_2)	12
3	3.0	20.08554 (y_3)	32
4	4.0	54.59815 (y_4)	7

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Boole's Rule Formula:

I = \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 \right]

Substituting values:

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

I = \frac{2}{45} \left[ 7.00000 + 86.98496 + 88.66872 + 642.73728 + 382.18705 \right]

I = \frac{2}{45} (1207.57801) = 53.67013

FINAL RESULT

Approximated Integral: I ≈ 53.6701

\boxed{\int_0^4 e^x dx \approx 53.6701} \quad (\text{Exact: } e^4 - 1 \approx 53.59815)

With $n = 4$ subintervals, Boole's Rule evaluates 53.6701, matching the true analytical exponential integral with remarkable precision!

LABORATORY

Interactive Live Calculator & Visualizer

FOURTH-ORDER INTEGRATION LABORATORY

Boole's Rule Engine

PreviousSimpson 3/8 Rule

Unit 3•15 of 28

NextWeddle's Rule

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL INTEGRATION

Boole's Rule
Quartic Integration

BOOLE'S CLOSED 4TH-ORDER FORMULAMULTIPLE OF 4 REQUIREMENT

I \approx \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 + \dots \right]

1. Pre-factor (2h / 45)2 times step size over 45:

h = \frac{b-a}{n}

2. Weight PatternBlock weights repeat in sequence: 7, 32, 12, 32, 7.

3. Shared BoundariesBlock boundaries sum:

7 + 7 = 14

at interior segment junctions.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the definite exponential integral:

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

Using $n = 4$ subintervals (multiple of 4)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 4$ and $n = 4$ :

h = \frac{b - a}{n} = \frac{4 - 0}{4} = 1.0

STEP 02 • TABULATE FUNCTION VALUES & BOOLE WEIGHTS

i	x_i	y_i = e^(x_i)	Boole Weight (w_i)
0	0.0	1.00000 (y_0)	7
1	1.0	2.71828 (y_1)	32
2	2.0	7.38906 (y_2)	12
3	3.0	20.08554 (y_3)	32
4	4.0	54.59815 (y_4)	7

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Boole's Rule Formula:

I = \frac{2h}{45} \left[ 7y_0 + 32y_1 + 12y_2 + 32y_3 + 7y_4 \right]

Substituting values:

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

I = \frac{2}{45} \left[ 7.00000 + 86.98496 + 88.66872 + 642.73728 + 382.18705 \right]

I = \frac{2}{45} (1207.57801) = 53.67013

FINAL RESULT

Approximated Integral: I ≈ 53.6701

\boxed{\int_0^4 e^x dx \approx 53.6701} \quad (\text{Exact: } e^4 - 1 \approx 53.59815)

With $n = 4$ subintervals, Boole's Rule evaluates 53.6701, matching the true analytical exponential integral with remarkable precision!

LABORATORY

Interactive Live Calculator & Visualizer

FOURTH-ORDER INTEGRATION LABORATORY

Boole's Rule Engine

PreviousSimpson 3/8 Rule

Unit 3•15 of 28

NextWeddle's Rule

Boole's Rule Quartic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 53.6701

Interactive Live Calculator & Visualizer

Boole's Rule Engine

NETZ Engine

Boole's Rule Quartic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 53.6701

Interactive Live Calculator & Visualizer

Boole's Rule Engine

Boole's Rule
Quartic Integration

Boole's Rule
Quartic Integration