Trapezoidal Rule Calculator – Numeric Integrals

The trapezoidal rule joins sample points with straight lines and sums the resulting trapezoids. For ∫₀¹ x² dx with n = 2 (h = 0.5, nodes 0, 0.5, 1): (0.5/2)[0 + 2(0.25) + 1] = 0.25 × 1.5 = 0.375, versus the exact 1/3 ≈ 0.33333 — a 12.5% overestimate because chords arch above a convex curve. Refining to n = 4 gives (0.25/2)[0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1] = 0.34375, error 3.1%: error scales as h², so doubling n quarters it. The calculator prints the node table (x, f(x), weight), the weighted sum, and the estimate, alongside the midpoint/Simpson values for comparison — Simpson takes the same nodes and fixes most of the curvature error. Concave functions flip the bias and underestimate instead.

1. 1

   Enter f(x) and bounds

   Any integrand over [a, b]

2. 2

   Choose n

   More intervals mean smaller h and less error

3. 3

   Build the node table

   Each xₖ, f(xₖ) and weight 2 (interior) or 1 (ends)

4. 4

   Sum and scale

   Multiply by h/2 for the estimate

• 1

  Error scales with h² — double the nodes, quarter the error

• 2

  Convex curves overestimate; concave curves underestimate

• 3

  Interior nodes weight 2; endpoints weight 1

• Weighting endpoints by 2 instead of 1

• Multiplying by h instead of h/2

• Expecting exactness on curves — only linear functions integrate exactly

• Comparing against Simpson without noting n must differ (Simpson needs even n)