Trapezoid Rule: Numerical Integration

What is the trapezoid rule?

The trapezoidal rule is a numerical method used to find the approximate area enclosed by a curve, the $x$ -axis and two vertical lines
- it is also known as ‘trapezoid rule’ and ‘trapezium rule’

The trapezoidal rule finds an approximation of the area by summing of the areas of trapezoids beneath the curve
- $y_{0} = f (a), y_{1} = f (a + h), y_{2} = f (a + 2 h)$ etc

\int_{a}^{b} f (x) d x \approx \frac{1}{2} h [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1})]

where

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

Note that there are $n$ trapezoids (also called strips) but $(n + 1)$ function values $(y_{i})$

The trapezoidal rule is given in the formula booklet

What else can I be asked to do with the trapezoid rule?

Comparing the true answer with the answer from the trapezoid rule
- This may involve finding the percentage error in the approximation
- The true answer may be given in the question, found from a GDC or from work on integration

Examiner Tip

Ensure you are clear about the difference between the number of data points ( $y$ values) and the number of strips (number of trapezoids) used in a Trapezoid Rule question
Although it shouldn't be too much trouble to type the trapezoid rule into your GDC in one go, it may be wise to work parts of it out separately and write these down as part of your working out

Worked example

Using the trapezoidal rule, find an approximate value for

\int_{0}^{4} \frac{6 x^{2}}{x^{3} + 2} d x

to 3 decimal places, using

n = 4

5-2-1-ib-si-ai-only-trap-rule-we-old-crop-a

Given that the area bounded by the curve , the

x

-axis and the lines

x = 0

and

x = 4

is 6.993 to three decimal places, calculate the percentage error in the trapezoidal rule approximation.

5-2-1-ib-si-ai-only-trap-rule-we-old-crop-b

Trapezoid Rule: Numerical Integration (DP IB Maths: AI SL)

Revision Note

What is the trapezoid rule?

What else can I be asked to do with the trapezoid rule?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Exponents & Logarithms

1.1.3 Approximation & Estimation

1.1.4 GDC: Solving Equations

1.2 Sequences & Series

1.2.1 Language of Sequences & Series

1.2.2 Arithmetic Sequences & Series

1.2.3 Geometric Sequences & Series

1.2.4 Applications of Sequences & Series

1.3 Financial Applications

1.3.1 Compound Interest & Depreciation

1.3.2 Amortisation & Annuities

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear & Piecewise Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Voronoi Diagrams

3.4.1 Voronoi Diagrams

3.4.2 Toxic Waste Dump Problem

4. Statistics & Probability

4.1 Statistics Toolkit

4.1.1 Sampling & Data Collection

4.1.2 Statistical Measures

4.1.3 Frequency Tables

4.1.4 Linear Transformations of Data

4.1.5 Outliers

4.1.6 Univariate Data

4.1.7 Interpreting Data

4.2 Correlation & Regression

4.2.1 Bivariate data

4.2.2 Correlation Coefficients

4.2.3 Linear Regression

4.3 Probability

4.3.1 Probability & Types of Events

4.3.2 Conditional Probability

4.3.3 Sample Space Diagrams

4.4 Probability Distributions

4.4.1 Discrete Probability Distributions

4.4.2 Expected Values

4.5 Binomial Distribution

4.5.1 The Binomial Distribution

4.5.2 Calculating Binomial Probabilities

4.6 Normal Distribution

4.6.1 The Normal Distribution

4.6.2 Calculations with Normal Distribution

4.7 Hypothesis Testing

4.7.1 Hypothesis Testing

4.7.2 Chi-squared Test for Independence

4.7.3 Goodness of Fit Test

4.7.4 The t-test