Trapezoidal Rule Calculator

The Trapezoidal Rule Calculator estimates a definite integral by replacing parts of a curve with trapezoids. It is especially useful when an exact antiderivative is difficult to obtain or when values are available only at sample points. Use the tool to work through the calculation and review the result.

Definite
Indefinite

with respect to

Upper value

Lower value

Trapezoidal Rule formula

With equally spaced points and step size h, the composite rule is T = h/2 [f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)].

Worked example

Estimate ∫₀² x² dx using two subintervals. Here h = 1 and the function values are f(0)=0, f(1)=1, and f(2)=4. The estimate is 1/2[0 + 2(1) + 4] = 3. The exact integral is 8/3, so the trapezoidal estimate is close but not exact.

Improving the estimate

Using more, smaller subintervals generally improves the approximation for a smooth curve. The calculator can make this repeated calculation much faster.

Frequently Asked Questions FAQ

What does the Trapezoidal Rule approximate?

It estimates a definite integral by replacing sections of the curve with trapezoids and adding their areas.

What happens when more subintervals are used?

For a smooth function, smaller subintervals generally produce a better approximation because the straight-line segments follow the curve more closely.

What is a simple numerical check?

For f(x)=x² on [0,2] with two subintervals, the trapezoidal estimate is 3, compared with the exact integral 8/3.

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