Trapezium Rule

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math

Approximates the area of a curve over a region by splitting the area into a number of connected trapeziums. This is often used as an alternative to integration.

For example consider a simple straight line curve with no gradient \( y = 5 \). If we want to find the area of the curve we can set \( y = a \) (the start position), step \( a \) until we reach \( b \) and then sum the area of the regions we stepped through together. Each region can be thought of as a trapezium, hence the trapezium rule.

\begin{align} \int^b_a y \,dx \approx \frac{1}{2} h \{ (y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1}) \}, \quad h = \frac{b - a}{n} = \frac{\text{lowest \( x \) - \text{smallest \( x \)}}}{\text{number of sides}} \end{align}