UNIT 3 • NUMERICAL INTEGRATION

Trapezoidal Rule
Numerical Integration

The Trapezoidal Rule approximates a definite integral $\int_a^b f(x) dx$ by connecting neighboring function evaluations with straight lines, forming a series of trapezoids whose areas sum to the total definite integral!

COMPOSITE TRAPEZOIDAL FORMULALINEAR INTERPOLATION

I = \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \right]

1. Subinterval Step (h)

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

(width of each trapezoid strip).

2. End Heights (y₀ + yₙ)The boundaries at

a

and

b

are counted once.

3. Interior Heights (2 × Sum)Shared internal boundaries are counted twice:

2(y_1 + \dots + y_{n-1})

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the classic definite integral:

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

Using $n = 6$ subintervals!

STEP 01 • CALCULATE STRIP STEP SIZE (h)

With integration limits $a = 0, b = 1$ and $n = 6$ subintervals:

h = \frac{b - a}{n} = \frac{1 - 0}{6} = \frac{1}{6} \approx 0.166667

STEP 02 • TABULATE GRID POINTS & FUNCTION VALUES

i	x_i	f(x_i) = 1 / (1 + x_i²)	Category
0	0.00000	1.00000 (y_0)	Boundary End
1	0.16667	0.97297 (y_1)	Interior (×2)
2	0.33333	0.90000 (y_2)	Interior (×2)
3	0.50000	0.80000 (y_3)	Interior (×2)
4	0.66667	0.69231 (y_4)	Interior (×2)
5	0.83333	0.59016 (y_5)	Interior (×2)
6	1.00000	0.50000 (y_6)	Boundary End

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Trapezoidal Integration Formula:

I = \frac{h}{2} \left[ (y_0 + y_6) + 2(y_1 + y_2 + y_3 + y_4 + y_5) \right]

Substituting values:

I = \frac{1/6}{2} \left[ (1.00000 + 0.50000) + 2(0.97297 + 0.90000 + 0.80000 + 0.69231 + 0.59016) \right]

I = \frac{1}{12} \left[ 1.50000 + 2(3.95544) \right] = \frac{1}{12} \left[ 1.50000 + 7.91088 \right]

I = \frac{1}{12} (9.41088) = 0.78424

FINAL RESULT

Approximated Integral: I ≈ 0.78424

\boxed{\int_0^1 \frac{1}{1+x^2} dx \approx 0.78424} \quad (\text{Exact: } \frac{\pi}{4} \approx 0.78540)

With $n = 6$ trapezoidal strips, the numerical integral 0.78424 matches the analytical value $\pi/4$ to 3 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

NUMERICAL INTEGRATION LABORATORY

Trapezoidal Rule Interactive Engine

PreviousNewton Divided Difference

Unit 3•12 of 28

NextSimpson 1/3 Rule

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL INTEGRATION

Trapezoidal Rule
Numerical Integration

COMPOSITE TRAPEZOIDAL FORMULALINEAR INTERPOLATION

I = \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \right]

1. Subinterval Step (h)

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

(width of each trapezoid strip).

2. End Heights (y₀ + yₙ)The boundaries at

a

and

b

are counted once.

3. Interior Heights (2 × Sum)Shared internal boundaries are counted twice:

2(y_1 + \dots + y_{n-1})

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Evaluate the classic definite integral:

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

Using $n = 6$ subintervals!

STEP 01 • CALCULATE STRIP STEP SIZE (h)

With integration limits $a = 0, b = 1$ and $n = 6$ subintervals:

h = \frac{b - a}{n} = \frac{1 - 0}{6} = \frac{1}{6} \approx 0.166667

STEP 02 • TABULATE GRID POINTS & FUNCTION VALUES

i	x_i	f(x_i) = 1 / (1 + x_i²)	Category
0	0.00000	1.00000 (y_0)	Boundary End
1	0.16667	0.97297 (y_1)	Interior (×2)
2	0.33333	0.90000 (y_2)	Interior (×2)
3	0.50000	0.80000 (y_3)	Interior (×2)
4	0.66667	0.69231 (y_4)	Interior (×2)
5	0.83333	0.59016 (y_5)	Interior (×2)
6	1.00000	0.50000 (y_6)	Boundary End

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Trapezoidal Integration Formula:

I = \frac{h}{2} \left[ (y_0 + y_6) + 2(y_1 + y_2 + y_3 + y_4 + y_5) \right]

Substituting values:

I = \frac{1/6}{2} \left[ (1.00000 + 0.50000) + 2(0.97297 + 0.90000 + 0.80000 + 0.69231 + 0.59016) \right]

I = \frac{1}{12} \left[ 1.50000 + 2(3.95544) \right] = \frac{1}{12} \left[ 1.50000 + 7.91088 \right]

I = \frac{1}{12} (9.41088) = 0.78424

FINAL RESULT

Approximated Integral: I ≈ 0.78424

\boxed{\int_0^1 \frac{1}{1+x^2} dx \approx 0.78424} \quad (\text{Exact: } \frac{\pi}{4} \approx 0.78540)

With $n = 6$ trapezoidal strips, the numerical integral 0.78424 matches the analytical value $\pi/4$ to 3 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

NUMERICAL INTEGRATION LABORATORY

Trapezoidal Rule Interactive Engine

PreviousNewton Divided Difference

Unit 3•12 of 28

NextSimpson 1/3 Rule

Trapezoidal Rule Numerical Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 0.78424

Interactive Live Calculator & Visualizer

Trapezoidal Rule Interactive Engine

NETZ Engine

Trapezoidal Rule Numerical Integration

Step-by-Step Worked Example

Approximated Integral: I ≈ 0.78424

Interactive Live Calculator & Visualizer

Trapezoidal Rule Interactive Engine

Trapezoidal Rule
Numerical Integration

Trapezoidal Rule
Numerical Integration