What is the trapezoidal rule?
A numerical integration method that approximates the area under a curve with trapezoids.
Trapezoidal Rule Calculator
Use equal-width trapezoids on [a, b] for f(x)=x², or integrate a list of y-values with spacing h.
Choose data source
Select f(x)=x² with a, b, n — or paste ordered y-values with step h.
About this calculator
How to use Trapezoidal Rule Calculator, plus the formulas and logic behind the results.
Tool description
The trapezoidal rule averages left and right Riemann sums, equivalent to connecting sample points with straight segments.
How to use
- Choose f(x)=x² mode or custom y-values.
- Enter interval and n, or h and the y list.
- Select Calculate to view the approximation.
- Increase n (or sample density) to refine the estimate.
Formulas & reference
T = (Δx/2) [y₀ + 2y₁ + … + 2yₙ₋₁ + yₙ]Δx = (b − a)/n (equal spacing)
Trapezoid tips
- Endpoints get weight 1; interior points get weight 2
- y-list mode needs equally spaced samples
- Compare with the exact ∫ x² dx = (b³−a³)/3
- n must be a positive integer in x² mode
Calculation logic
Samples f at equal nodes or uses provided y-values, then applies the composite trapezoidal weights.
Requires at least two sample points. Error decreases like O(1/n²) for smooth f.
Trapezoidal Rule Calculator FAQ
Common questions about this calculator and how to use the results.
When use y-list mode?
When you already have tabulated function values at equal spacing h.
How many y-values?
At least two (the two endpoints of one trapezoid).
Is it more accurate than left Riemann?
Usually yes for smooth functions, similar to averaging left and right sums.
Must spacing be equal?
Yes for this composite formula.
Is this free?
Yes.