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

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

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