Trapezium Rule (Numerical Integration)

• The method relies on linear interpolation, assuming that over a very small interval, a curve can be reasonably approximated by a straight line.

• The area of a single trapezium with parallel vertical sides $y_i$ and $y_{i+1}$ and width $h$ is given by $Area = \frac{1}{2}h(y_i + y_{i+1})$.

• When summing $n$ trapeziums, the 'inner' boundaries are shared by two adjacent trapeziums, meaning those heights are counted twice in the total sum.

• The 'outer' boundaries (the first and last heights) are only part of one trapezium each, so they are only counted once.

• Step 1: Determine the strip width. Calculate $h = \frac{b - a}{n}$. Ensure $n$ is the number of strips, not the number of points.

• Step 2: Generate the x-values. Start at $x_0 = a$ and add $h$ repeatedly until you reach $x_n = b$.

• Step 3: Calculate the y-values (ordinates). Substitute each $x$-value into the function $f(x)$ to find $y_0, y_1, \dots, y_n$. It is best to organize these in a table.

• Step 4: Apply the formula. Substitute the values into the general rule:

$\int_a^b y \, dx \approx \frac{1}{2}h \left[ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \right]$

• Increasing Strips: As the number of strips $n$ increases, the width $h$ decreases, and the straight-line approximation becomes closer to the actual curve, reducing the error.

• Concavity Impact: The direction of the error depends on the curvature of the function. If the curve is concave down ($f''(x) < 0$), the rule produces an underestimate. If the curve is concave up ($f''(x) > 0$), it produces an overestimate.

• Percentage Error: Accuracy is often measured by comparing the estimate to the exact value (if known) using the formula: $Percentage \, Error = \frac{|Estimate - Exact|}{Exact} \times 100$.

• It is important to distinguish between strips ($n$) and ordinates ($n+1$). For example, 5 strips require 6 $y$-values.

• Check your Calculator Mode: If the function involves trigonometry, ensure your calculator is in Radians unless specified otherwise.

• Rounding Precision: Keep intermediate $y$-values to a higher degree of accuracy (e.g., 5 or 6 decimal places) than the final answer requires to avoid rounding accumulation errors.

• The '2' Factor: A common mistake is forgetting that only the middle ordinates are multiplied by 2. Always double-check the brackets in your formula.

• Sanity Check: Look at the graph of the function. If the curve is clearly bending upwards, your answer should be slightly larger than the true area.