The trapezoidal rule approximates the area under a curve (or between data points) by summing trapezoids of equal width. It’s useful when the exact integral is difficult, or when you have tabulated data instead of a function.
| Term / Concept | Definition / Short explanation |
|---|---|
| Trapezoidal rule (n subintervals) | For interval [a,b] split into n equal parts of width h=(b−a)/n, area ≈ h[ (1/2)f(a) + f(x1) + … + f(xn−1) + (1/2)f(b) ]. |
| Equal width requirement | In SL use, intervals are usually equal width; if data spacing is unequal, apply trapezoid formula piecewise with each width. |
| Error note | The trapezoidal rule is exact for linear functions; error depends on curvature (second derivative). For smooth functions, refine (increase n) to reduce error. |
📌 How to apply the trapezoidal rule
- Identify interval [a,b] and number of subintervals n (or use data points given).
- Compute width h = (b − a)/n.
- Evaluate f at the endpoints and interior points (or use given table values).
- Compute area ≈ h × [ (1/2)f(a) + Σ f(xi) (i=1..n−1) + (1/2)f(b) ].
- If data spacing is irregular, compute each trapezoid area individually: (width)k × (f(left)+f(right))/2 and sum.
Worked example — trapezoidal rule
Approximate the area under f(x) = x3 on [0, 2] using the trapezoidal rule with n = 4.
- Find h: h = (2 − 0)/4 = 0.5.
- Compute f at nodes: xi = 0.0, 0.5, 1.0, 1.5, 2.0 → f = 0, 0.125, 1, 3.375, 8.
- Apply trapezoidal formula:
Area ≈ h · [ (1/2)f(x0) + f(x1) + f(x2) + f(x3) + (1/2)f(x4) ]
= 0.5 · [ 0/2 + 0.125 + 1 + 3.375 + 8/2 ] = 0.5 · 8.50 = 4.25.
- Exact integral (for comparison): ∫02 x3 dx = (1/4)x4|02 = 4. Error = 0.25 (overestimate).
🧠 Tip
Keep the node table short and present h → node values → bracket sum → multiply by h. Use more subintervals (larger n) to reduce error.
🧠 Paper tip
- Write h explicitly and list the function values used — examiners expect the formula and the substitution shown.
- If given a table, verify whether spacing is equal; if not, compute trapezoid areas individually and state why.
🌍 Real-world connection
Approximating areas is used in environmental science — estimating lake area from survey cross-sections — and in physics to estimate displacement from discrete velocity-time measurements when an analytic velocity function is unavailable.
GDC tip: Many calculators and graphing tools have a numeric integral or trapezoid function — when using technology show the input (a, b, n) and state the numeric result; still show the formula on paper for method marks.
🔗 Connections
- Physics: displacement ≈ area under v(t) from sampled velocity data.
- Numerical methods: trapezoidal rule is a basic quadrature technique — compare with Simpson’s rule to discuss accuracy.
- Paper tip (connected): when asked to estimate area from a table always state whether you used equal-width trapezoids or handled unequal widths individually.
📝 Final examiner note
Keep answers concise: show the trapezoidal formula, the h value, the list of f(x) used, the arithmetic, and a short interpretation of the approximation (mention whether more subintervals would increase accuracy).