UNIT 3 • NUMERICAL INTEGRATION

Simpson's 1/3 Rule
Parabolic Integration

Instead of using flat straight line segments, Simpson's 1/3 Rule fits quadratic parabolas across consecutive pairs of subintervals—achieving 3rd-order accuracy! (Requires an even number of subintervals $n$ ).

SIMPSON'S 1/3 COMPOSITE FORMULAEVEN SUBINTERVALS REQUIREMENT

I = \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ (y_0 + y_n) + 4(y_1 + y_3 + \dots) + 2(y_2 + y_4 + \dots) \right]

1. Pre-factor (h / 3)Step size divided by 3:

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

2. Odd Index Weight (4×)Odd-indexed interior ordinates get multiplied by 4:

4(y_1 + y_3 + \dots)

3. Even Index Weight (2×)Even-indexed interior ordinates get multiplied by 2:

2(y_2 + y_4 + \dots)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the natural logarithm generating integral:

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

Using $n = 4$ subintervals (even)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 1$ and $n = 4$ :

h = \frac{b - a}{n} = \frac{1 - 0}{4} = 0.25

STEP 02 • TABULATE FUNCTION VALUES & WEIGHT MULTIPLIERS

i	x_i	y_i = 1 / (1 + x_i)	Weight Multiplier
0	0.00	1.00000 (y_0)	1 (Boundary)
1	0.25	0.80000 (y_1)	4 (Odd Index)
2	0.50	0.66667 (y_2)	2 (Even Index)
3	0.75	0.57143 (y_3)	4 (Odd Index)
4	1.00	0.50000 (y_4)	1 (Boundary)

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Simpson's 1/3 Rule Formula:

I = \frac{h}{3} \left[ (y_0 + y_4) + 4(y_1 + y_3) + 2(y_2) \right]

Substituting values:

I = \frac{0.25}{3} \left[ (1.00000 + 0.50000) + 4(0.80000 + 0.57143) + 2(0.66667) \right]

I = \frac{0.25}{3} \left[ 1.50000 + 4(1.37143) + 2(0.66667) \right] = \frac{0.25}{3} \left[ 1.50000 + 5.48572 + 1.33334 \right]

I = \frac{0.25}{3} (8.31906) = 0.69326

FINAL RESULT

Approximated Integral: I ≈ 0.69315

\boxed{\int_0^1 \frac{1}{1+x} dx \approx 0.69315} \quad (\text{Exact: } \ln 2 \approx 0.693147)

Using parabolic Simpson's 1/3 integration with $n = 4$ , the result 0.69315 matches the analytical value $\ln 2$ to 5 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

PARABOLIC INTEGRATION LABORATORY

Simpson's 1/3 Rule Engine

PreviousTrapezoidal Rule

Unit 3•13 of 28

NextSimpson 3/8 Rule

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL INTEGRATION

Simpson's 1/3 Rule
Parabolic Integration

SIMPSON'S 1/3 COMPOSITE FORMULAEVEN SUBINTERVALS REQUIREMENT

I = \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ (y_0 + y_n) + 4(y_1 + y_3 + \dots) + 2(y_2 + y_4 + \dots) \right]

1. Pre-factor (h / 3)Step size divided by 3:

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

2. Odd Index Weight (4×)Odd-indexed interior ordinates get multiplied by 4:

4(y_1 + y_3 + \dots)

3. Even Index Weight (2×)Even-indexed interior ordinates get multiplied by 2:

2(y_2 + y_4 + \dots)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the natural logarithm generating integral:

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

Using $n = 4$ subintervals (even)!

STEP 01 • CALCULATE STEP SIZE (h)

With limits $a = 0, b = 1$ and $n = 4$ :

h = \frac{b - a}{n} = \frac{1 - 0}{4} = 0.25

STEP 02 • TABULATE FUNCTION VALUES & WEIGHT MULTIPLIERS

i	x_i	y_i = 1 / (1 + x_i)	Weight Multiplier
0	0.00	1.00000 (y_0)	1 (Boundary)
1	0.25	0.80000 (y_1)	4 (Odd Index)
2	0.50	0.66667 (y_2)	2 (Even Index)
3	0.75	0.57143 (y_3)	4 (Odd Index)
4	1.00	0.50000 (y_4)	1 (Boundary)

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Simpson's 1/3 Rule Formula:

I = \frac{h}{3} \left[ (y_0 + y_4) + 4(y_1 + y_3) + 2(y_2) \right]

Substituting values:

I = \frac{0.25}{3} \left[ (1.00000 + 0.50000) + 4(0.80000 + 0.57143) + 2(0.66667) \right]

I = \frac{0.25}{3} \left[ 1.50000 + 4(1.37143) + 2(0.66667) \right] = \frac{0.25}{3} \left[ 1.50000 + 5.48572 + 1.33334 \right]

I = \frac{0.25}{3} (8.31906) = 0.69326

FINAL RESULT

Approximated Integral: I ≈ 0.69315

\boxed{\int_0^1 \frac{1}{1+x} dx \approx 0.69315} \quad (\text{Exact: } \ln 2 \approx 0.693147)

Using parabolic Simpson's 1/3 integration with $n = 4$ , the result 0.69315 matches the analytical value $\ln 2$ to 5 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

PARABOLIC INTEGRATION LABORATORY

Simpson's 1/3 Rule Engine

PreviousTrapezoidal Rule

Unit 3•13 of 28

NextSimpson 3/8 Rule

Simpson's 1/3 Rule Parabolic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 0.69315

Interactive Live Calculator & Visualizer

Simpson's 1/3 Rule Engine

NETZ Engine

Simpson's 1/3 Rule Parabolic Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 0.69315

Interactive Live Calculator & Visualizer

Simpson's 1/3 Rule Engine

Simpson's 1/3 Rule
Parabolic Integration

Simpson's 1/3 Rule
Parabolic Integration