Trapezoidal Rule

\int_{a}^{b} f (x) d x \approx \frac{Δ x}{2} (y_{0} + 2 i = 1 \sum n - 1 y_{i} + y_{n}), Δ x = \frac{b - a}{n}

The trapezoidal rule approximates the definite integral by n trapezoids of width Δx. End ordinates are counted once; interior ordinates twice. Error is typically larger than Simpson's rule (which needs even n).

$a, b$: Limits of integration
$n$: Number of subintervals
$Δ x$: Width of each panel
$y_{i}$: f at the i-th node

Step-by-step solved example

Approximate ∫_0^2 x² dx with n = 2 trapezoids.

Trapezoidal rule: piecewise linear area under the curve.

1. Spacing
Δx = (2−0)/2 = 1. Nodes x = 0, 1, 2 → y = 0, 1, 4.
2. Rule
(Δx/2)[y0 + 2y1 + y2] = 0.5[0 + 2 + 4] = 3.
$\approx 3$
3. Exact check
Exact = 8/3 ≈ 2.667. Trapezoids overestimate a convex-up parabola.

Answer: 3 (exact 8/3)

10 practice questions

0/10 correct

1.n = 4 from 0 to 2. Δx equals

2.Interior ordinates in the trapezoidal formula are multiplied by

3.∫_0^1 1 dx with n = 1 trapezoid equals

4.Simpson's 1/3 rule, unlike trapezoidal, requires

5.y = [1, 2, 3], Δx = 1. Trapezoidal integral ≈

6.Increasing n generally

7.The trapezoidal rule is exact for

8.Composite formula: (Δx/2)(y0 + yn + 2 Σ interior). For n panels there are

9.f(0)=0, f(2)=4, f(4)=0, Δx=2. Integral ≈

10.Units of the result are