Trapezoidal Rule Calculator

The Trapezoidal Rule approximates a definite integral by dividing the area under the curve into trapezoids rather than rectangles. Each trapezoid connects adjacent function values with a straight line. The rule has second-order accuracy — halving the step size reduces the error by a factor of four.

1. 1T = (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
2. 2h = (b−a)/n is the step size
3. 3Error ≈ −(b−a)³/(12n²) × f''(ξ) for some ξ in [a,b]
4. 4Error is zero when f is linear (trapezoids fit exactly)
5. 5Simpson's Rule corrects the trapezoid error using parabolic interpolation