Trapezoidal rule

Trapezoidal Rule is a numerical method used in mathematics for approximating the definite integral of a function. It is a type of Numerical integration that works by dividing the total area under a curve into trapezoids, rather than rectangles (as in the Rectangle method), and then summing the areas of these trapezoids to approximate the integral. The trapezoidal rule is particularly useful when an analytic expression of the integral is difficult to obtain, or when dealing with empirical data.

Overview

The basic idea behind the trapezoidal rule is to approximate the area under a curve between two points, \(a\) and \(b\), by calculating the area of the trapezoid formed by the linear line connecting these points. This method assumes that the function \(f(x)\) to be integrated is continuous and well-behaved in the interval \([a, b]\).

Formula

The formula for the trapezoidal rule for a single interval from \(a\) to \(b\) is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2} [f(a) + f(b)] \] For multiple intervals, the interval \([a, b]\) is divided into \(n\) subintervals of equal length \(h = \frac{b-a}{n}\). The approximation of the integral over these \(n\) intervals is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)] \] where \(x_0 = a\), \(x_n = b\), and \(x_i = a + ih\) for \(i = 1, 2, \ldots, n-1\).

Application

The trapezoidal rule is widely used in various fields such as physics, engineering, and economics for solving problems that require the calculation of definite integrals. Its simplicity and ease of implementation make it a popular choice for introductory numerical analysis courses.

Accuracy

The accuracy of the trapezoidal rule depends on the number of intervals \(n\) and the behavior of the function \(f(x)\). In general, increasing \(n\) improves the approximation, but at the cost of increased computational effort. The error in the trapezoidal rule can be estimated and is related to the second derivative of \(f(x)\) over the interval \([a, b]\).

Limitations

While the trapezoidal rule is effective for many practical applications, it may not provide accurate results for functions that are highly oscillatory or have discontinuities within the interval. In such cases, other methods of numerical integration, such as Simpson's rule or Gaussian quadrature, may be more appropriate.