Trapezoidal Rule Calculator

This calculator is designed for students, engineers, and mathematicians who need an accurate numerical approximation of definite integrals using the trapezoidal rule. Enter the function and interval details below to compute the integral.

Calculator

Function (f(x)) *

Lower Limit (a) *

Upper Limit (b) *

Number of Intervals (n) *

Results

Approximate Integral: 0

Data Source and Methodology

All calculations are strictly based on numerical methods for integration as described in mathematical literature.

The Formula Explained

\[ \text{Integral} \approx \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(a + i \cdot \Delta x) + f(b) \right) \]

Glossary of Terms

Function (f(x)): The mathematical expression to be integrated.
Lower Limit (a): The starting point of the integration interval.
Upper Limit (b): The ending point of the integration interval.
Number of Intervals (n): The number of subintervals used for the approximation.

How It Works: A Step-by-Step Example

Consider integrating f(x) = x^2 from a = 0 to b = 1 with n = 4 intervals. Substituting these values into the trapezoidal formula provides the approximate integral.

Frequently Asked Questions (FAQ)

What is the trapezoidal rule?

The trapezoidal rule is a numerical method used to approximate the definite integral of a function.

Why use numerical integration?

Numerical integration is used when an analytic integral is difficult or impossible to calculate.

How do I choose the number of intervals?

The number of intervals affects the accuracy; more intervals generally lead to better approximations.

Can this calculator handle discontinuous functions?

The trapezoidal rule may not be accurate for functions with discontinuities within the interval.

Is the trapezoidal rule always accurate?

While useful, the trapezoidal rule can have errors, especially for functions with significant curvature.

Tool developed by Ugo Candido. Content reviewed by experts.
Last reviewed for accuracy on: October 1, 2023.