Trapezoidal Rule (Leaving Cert Mathematics): Revision Notes

Trapezoidal Rule

What is the Trapezoidal Rule?

The Trapezoidal Rule is a method to approximate the area under a curve or the integral of a function. It works by dividing the area under the curve into a series of trapezoids and summing their areas. This is particularly useful for irregular shapes or when the function cannot be integrated easily.

Formula for the Trapezoidal Rule

For a curve described by a function $f(x)$ between $x = a$ and $x = b$, divided into n equal intervals:

$$\text{Area} \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right]$$

where:

• h = \frac{b - a}{n} is the width of each interval.
• $x_0, x_1, \ldots, x_n$ are the points of division.

A common way to remember this is: first + last + 2(sum of the rest), all multiplied by $\frac{h}{2}$.

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Steps to Use the Trapezoidal Rule

1. Divide the Interval: Split the range $[a, b]$ into n equal parts.
2. Calculate Heights: Evaluate the function $f(x)$ at each division point $x_0, x_1, \ldots, x_n$
3. Apply the Formula: Use the formula to approximate the area by summing the contributions of each trapezoid.

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Worked Examples

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Summary

• Purpose: Approximate areas under curves or integrals.
• Formula: $\text{Area} \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + \ldots + 2f(x_{n-1}) + f(x_n) \right]$
• Key Steps:
  1. Divide the range into equal parts.
  2. Compute function values at these points.
  3. Sum contributions using the trapezoidal formula.
• Suitable for approximations when exact integration is difficult.