Trapezium Rule (Numerical Integration)

• Linear Interpolation: The rule assumes that over a very small interval, a curve can be approximated by a straight line connecting two points $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$.

• Summation of Areas: The total area is the sum of the areas of individual trapeziums. The area of one trapezium is $\frac{1}{2}h(y_i + y_{i+1})$.

• The Composite Formula: When summed, the internal $y$-values (ordinates) appear in two adjacent trapeziums, while the end values ($y_0$ and $y_n$) appear only once.

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

• Step 1: Determine $h$: Calculate the strip width using $h = \frac{b-a}{n}$, where $a$ and $b$ are the limits of integration and $n$ is the number of strips.

• Step 2: Generate x-values: Create a sequence of $x$ values starting at $a$ and increasing by $h$ until reaching $b$ (e.g., $x_0, x_0+h, x_0+2h, \dots$).

• Step 3: Calculate y-values: Evaluate the function $f(x)$ at each of these $x$ values to find the corresponding ordinates $y_0, y_1, \dots, y_n$.

• Step 4: Apply the Formula: Group the first and last $y$-values together, sum the remaining 'middle' $y$-values and multiply them by 2, then apply the full formula.

• Strips vs. Ordinates: It is vital to distinguish between the number of intervals ($n$) and the number of data points ($n+1$). Using the wrong count will result in an incorrect $h$ value.

• Curvature and Error: The accuracy of the approximation depends entirely on the second derivative (concavity) of the function.

• Formula Memory: The Trapezium Rule formula is frequently omitted from standard formula booklets; students must memorize the structure: $\frac{1}{2}h[Ends + 2(Middles)]$.

• Rounding Precision: To avoid rounding errors in the final answer, keep $y$-values to a higher degree of accuracy (at least 4 or 5 decimal places) than the final requested answer.

• Check the $h$ calculation: Always verify that $x_n$ actually equals the upper limit $b$. If it doesn't, there is an error in the calculation of $h$ or the number of steps taken.

• Percentage Error: Exams often ask for the percentage error, calculated as $\frac{|Estimate - Exact|}{Exact} \times 100$. This requires finding the exact integral analytically if possible.

• Incorrect $n$: Students often confuse 'number of $x$ values' with 'number of strips'. If a question provides 5 $x$ values, there are only 4 strips ($n=4$).

• Radians vs. Degrees: When integrating trigonometric functions, calculators MUST be in Radian mode. Using degrees is a common error that leads to completely incorrect ordinate values.

• Forgetting the $\frac{1}{2}h$: A frequent mistake is forgetting to multiply the entire sum by the $\frac{1}{2}h$ factor at the end of the calculation.