5.3.1 Trapezoid Rule: Numerical Integration

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 $(y_{i})$

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

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

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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.

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IB DP Maths: AI HL

5.3.1 Trapezoid Rule: Numerical Integration