Trapezoidal Rule Calculator

Trapezoidal Rule Calculator

What Is the Trapezoidal Rule?

The trapezoidal rule is a method for estimating the area under a curve by dividing it into a series of trapezoids, rather than rectangles.

Mathematically, for a function f(x) over an interval [a, b] with n subintervals:

∫ab f(x) dx ≈ (h / 2) × [ f(a) + 2∑f(xi) + f(b) ]

Where h = (b – a) / n is the width of each subinterval.

How the Trapezoidal Rule Calculator Works

• Enter your function f(x), interval [a, b], and number of subintervals (n).
• View a step-by-step trapezoidal rule approximation.
• Calculate the error bound instantly.
• See results for the composite trapezoidal rule.

Step-by-Step Example

Example: Integrate f(x) = x² over [0, 2] using the trapezoidal rule with n = 4.

h = (2 - 0) / 4 = 0.5
f(0) = 0
f(0.5) = 0.25
f(1) = 1
f(1.5) = 2.25
f(2) = 4

Apply the formula:

T = (0.5 / 2) × [ f(0) + 2 × (0.25 + 1 + 2.25) + f(2) ] = 2.75

The exact value of the integral is 8 / 3 = 2.6667, so the error is approximately 0.0833.

Error Bound in the Trapezoidal Rule

The error bound formula is given by:

ET = -((b - a)3 / (12n2)) × f″(ξ)

Where ξ is a point in [a, b].

Composite Trapezoidal Rule

The composite trapezoidal rule divides [a, b] into several subintervals to improve accuracy. More subintervals give a result closer to the exact integral.

Trapezoidal Rule Formula Summary

T = (h / 2) × [ f(a) + 2∑f(xi) + f(b) ]

Applications

• Physics – Estimating displacement or work done.
• Engineering – Calculating power, energy, or charge.
• Finance – Estimating total returns over continuous curves.
• Computer Science – Numerical methods in simulations.

FAQs

Q1: What is the trapezoidal rule used for?
A: It’s used to estimate definite integrals when analytical solutions are difficult.

Q2: How accurate is it?
A: Accuracy increases as the number of subintervals (n) increases.

Q3: What’s the difference between Simpson’s and the trapezoidal rule?
A: Simpson’s rule fits parabolas instead of trapezoids, providing higher precision.

Q4: How do I calculate the error bound?
A: Use the formula E_T = ((b – a)³ / (12n²)) × f″(ξ).

Q5: Can this calculator handle tables of values?
A: Yes, enter discrete data points directly to estimate the integral.

Conclusion

The Trapezoidal Rule Calculator is a fast and accurate way to estimate integrals, calculate errors, and learn numerical methods. Try it below to see results instantly!