How to Calculate Trapezoidal Rule
What is Trapezoidal Rule?
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.
Formula
- T
- (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + — (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) +
Step-by-Step Guide
- 1T = (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
- 2h = (b−a)/n is the step size
- 3Error ≈ −(b−a)³/(12n²) × f''(ξ) for some ξ in [a,b]
- 4Error is zero when f is linear (trapezoids fit exactly)
- 5Simpson's Rule corrects the trapezoid error using parabolic interpolation
Worked Examples
Frequently Asked Questions
What is Trapezoidal Rule?
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
How accurate is the Trapezoidal Rule calculator?
The calculator uses the standard published formula for trapezoidal rule. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.
What units does the Trapezoidal Rule calculator use?
This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.
What formula does the Trapezoidal Rule calculator use?
The core formula is: T = (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]. Each step in the calculation is shown so you can verify the result manually.
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