Trapezoidal Quadrature: Composite Rules, Error, and Method Selection

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What does the single-interval trapezoidal rule approximate?

It approximates a definite integral by replacing the graph between two endpoints with a straight chord and using the resulting trapezoid's area.

What is the single-interval trapezoidal formula?

$\int_a^b f(x)\,dx \approx \frac{b-a}{2}\bigl(f(a)+f(b)\bigr)$.

How is the composite trapezoidal rule formed on a general partition?

Add the trapezoidal estimates for each subinterval, multiplying each average endpoint height by that subinterval's width.

When is the single-interval trapezoidal rule exact?

For a linear function, whose graph coincides with the chord joining its endpoint values.

What is the signed composite trapezoidal error on a uniform grid?

For sufficiently smooth $f$, $E=\int_a^b f(x)\,dx-T_N=-\frac{(b-a)^3}{12N^2}f''(\xi)$ for some $\xi\in(a,b)$.

Quiz questions

1. For a uniform grid with spacing $h$, how are the function values weighted in the compact composite trapezoidal formula?Answer: Interior values have weight $h$; the two endpoint values each have weight $h/2$.
2. If $f''(x)>0$ throughout an interval, how does the trapezoidal estimate compare with the exact integral?Answer: It overestimates the integral because the chords lie above the graph.
3. Under the composite error bound, what happens to the bound when the number of uniform subintervals is doubled?Answer: The bound becomes one quarter as large, with the interval and derivative bound fixed.
4. Why can the trapezoidal rule converge especially rapidly for a smooth periodic function integrated over a full period?Answer: Matching endpoint derivatives cancel contributions in the Euler–Maclaurin analysis.
5. For an infinitely smooth, non-periodic integrand, which methods can often achieve greater precision than the trapezoidal rule with the same number of evaluations?Answer: Gaussian or Clenshaw–Curtis quadrature.