AP Calculus AB and BC glossary

Trapezoidal rule

Also called: Trapezoid rule

The trapezoidal rule approximates a definite integral by joining consecutive points with straight segments and summing the resulting trapezoids. It equals the average of the left and right Riemann sums and is usually more accurate than either.

∫abf(x) dx≈Δx2[f(x0)+2f(x1)+⋯+2f(xn−1)+f(xn)]\int_a^b f(x)\,dx \approx \frac{\Delta x}{2}\left[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\right]

The doubled interior coefficients are not arbitrary: every interior point is shared by two adjacent trapezoids, so it gets counted twice while the two endpoints are counted once.

Concavity fixes the direction of the error. On a concave up curve the chords lie above the graph, so the trapezoidal rule overestimates; on a concave down curve it underestimates.

Unequal widths

The compact formula assumes equal spacing. With data at uneven intervals, compute each trapezoid separately as the average height times its own width.

Appears in: Unit 6: Integration and Accumulation