UNIT 3 • NUMERICAL INTEGRATION

Simpson's 3/8 Rule
Cubic Integration

Simpson's 3/8 Rule fits cubic polynomials across sets of 4 consecutive points (3 subintervals), providing exceptional accuracy when the total subintervals $n$ is a multiple of 3!

SIMPSON'S 3/8 COMPOSITE FORMULAMULTIPLE OF 3 REQUIREMENT

I = \int_{a}^{b} f(x) \, dx \approx \frac{3h}{8} \left[ (y_0 + y_n) + 3(y_1 + y_2 + y_4 + y_5 + \dots) + 2(y_3 + y_6 + \dots) \right]

1. Pre-factor (3h / 8)3 times step size over 8:

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

2. Non-Multiple 3 (3×)Indices not divisible by 3 get multiplied by 3:

3(y_1 + y_2 + y_4 + \dots)

3. Multiple of 3 (2×)Interior indices divisible by 3 get multiplied by 2:

2(y_3 + y_6 + \dots)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the exponential growth integral:

\int_{0}^{3} e^x \, dx

Using $n = 3$ subintervals (multiple of 3)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 3$ and $n = 3$ :

h = \frac{b - a}{n} = \frac{3 - 0}{3} = 1.0

STEP 02 • TABULATE FUNCTION VALUES & WEIGHT MULTIPLIERS

i	x_i	y_i = e^(x_i)	Weight Multiplier
0	0.0	1.00000 (y_0)	1 (Boundary)
1	1.0	2.71828 (y_1)	3 (Non-mult of 3)
2	2.0	7.38906 (y_2)	3 (Non-mult of 3)
3	3.0	20.08554 (y_3)	1 (Boundary)

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Simpson's 3/8 Rule Formula:

I = \frac{3h}{8} \left[ (y_0 + y_3) + 3(y_1 + y_2) \right]

Substituting values:

I = \frac{3(1.0)}{8} \left[ (1.00000 + 20.08554) + 3(2.71828 + 7.38906) \right]

I = 0.37500 \left[ 21.08554 + 3(10.10734) \right] = 0.37500 \left[ 21.08554 + 30.32202 \right]

I = 0.37500 (51.40756) = 19.27784

FINAL RESULT

Approximated Integral: I ≈ 19.2778

\boxed{\int_0^3 e^x dx \approx 19.2778} \quad (\text{Exact: } e^3 - 1 \approx 19.0855)

With $n = 3$ cubic subintervals, Simpson's 3/8 Rule computes 19.2778, accurately capturing steep exponential curvature!

LABORATORY

Interactive Live Calculator & Visualizer

CUBIC INTEGRATION LABORATORY

Simpson's 3/8 Rule Engine

PreviousSimpson 1/3 Rule

Unit 3•14 of 28

NextBoole's Rule

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL INTEGRATION

Simpson's 3/8 Rule
Cubic Integration

Simpson's 3/8 Rule fits cubic polynomials across sets of 4 consecutive points (3 subintervals), providing exceptional accuracy when the total subintervals $n$ is a multiple of 3!

SIMPSON'S 3/8 COMPOSITE FORMULAMULTIPLE OF 3 REQUIREMENT

I = \int_{a}^{b} f(x) \, dx \approx \frac{3h}{8} \left[ (y_0 + y_n) + 3(y_1 + y_2 + y_4 + y_5 + \dots) + 2(y_3 + y_6 + \dots) \right]

1. Pre-factor (3h / 8)3 times step size over 8:

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

2. Non-Multiple 3 (3×)Indices not divisible by 3 get multiplied by 3:

3(y_1 + y_2 + y_4 + \dots)

3. Multiple of 3 (2×)Interior indices divisible by 3 get multiplied by 2:

2(y_3 + y_6 + \dots)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the exponential growth integral:

\int_{0}^{3} e^x \, dx

Using $n = 3$ subintervals (multiple of 3)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 3$ and $n = 3$ :

h = \frac{b - a}{n} = \frac{3 - 0}{3} = 1.0

STEP 02 • TABULATE FUNCTION VALUES & WEIGHT MULTIPLIERS

i	x_i	y_i = e^(x_i)	Weight Multiplier
0	0.0	1.00000 (y_0)	1 (Boundary)
1	1.0	2.71828 (y_1)	3 (Non-mult of 3)
2	2.0	7.38906 (y_2)	3 (Non-mult of 3)
3	3.0	20.08554 (y_3)	1 (Boundary)

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Simpson's 3/8 Rule Formula:

I = \frac{3h}{8} \left[ (y_0 + y_3) + 3(y_1 + y_2) \right]

Substituting values:

I = \frac{3(1.0)}{8} \left[ (1.00000 + 20.08554) + 3(2.71828 + 7.38906) \right]

I = 0.37500 \left[ 21.08554 + 3(10.10734) \right] = 0.37500 \left[ 21.08554 + 30.32202 \right]

I = 0.37500 (51.40756) = 19.27784

FINAL RESULT

Approximated Integral: I ≈ 19.2778

\boxed{\int_0^3 e^x dx \approx 19.2778} \quad (\text{Exact: } e^3 - 1 \approx 19.0855)

With $n = 3$ cubic subintervals, Simpson's 3/8 Rule computes 19.2778, accurately capturing steep exponential curvature!

LABORATORY

Interactive Live Calculator & Visualizer

CUBIC INTEGRATION LABORATORY

Simpson's 3/8 Rule Engine

PreviousSimpson 1/3 Rule

Unit 3•14 of 28

NextBoole's Rule

Simpson's 3/8 Rule Cubic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 19.2778

Interactive Live Calculator & Visualizer

Simpson's 3/8 Rule Engine

NETZ Engine

Simpson's 3/8 Rule Cubic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 19.2778

Interactive Live Calculator & Visualizer

Simpson's 3/8 Rule Engine

Simpson's 3/8 Rule
Cubic Integration

Simpson's 3/8 Rule
Cubic Integration