Understanding and Applying the Trapezoidal Rule for Numerical Integration

The trapezoidal rule is a numerical method used to estimate the definite integral of a function, especially when the function is difficult to integrate analytically. This method approximates the area under a curve by dividing it into smaller trapezoids, offering a balance between simplicity and accuracy.

How It Works

Partitioning the Interval

The interval [a, b] over which you want to integrate the function f(x) is divided into n subintervals of equal width h, where:

h frac{b - a}{n}

Calculating the Trapezoids

For each subinterval [x_i, x_{i 1}] where x_i a ih for i 0, 1, ldots, n, the area of the trapezoid is calculated as:

text{Area}_i frac{h}{2} left [f(x_i) f(x_{i 1}) right]

Summing the Areas

The total area under the curve is approximated by summing the areas of all the trapezoids:

text{Integral} approx sum_{i0}^{n-1} text{Area}_i frac{h}{2} left [f(a) 2 sum_{i1}^{n-1} f(x_i) f(b) right]

Advantages and Disadvantages

Advantages

Simple to Understand and Implement: The trapezoidal rule is straightforward, making it easy for beginners to grasp and apply. Better Approximation for Linear Functions: The method provides a more accurate approximation than the rectangle method, especially for linear functions.

Disadvantages

Less Accurate for Highly Non-Linear Functions: The accuracy of the trapezoidal rule decreases for highly non-linear functions unless a large number of subintervals is used. Less Efficient Than Other Numerical Methods: For certain types of functions, the trapezoidal rule can be less efficient than other numerical integration methods, such as Simpson's rule.

Example

To integrate the function f(x) x^2 from 0 to 2 using the trapezoidal rule with n 4:

a 0 b 2 h frac{2 - 0}{4} 0.5

Calculate the points:

x_0 0 x_1 0.5 x_2 1 x_3 1.5 x_4 2

Evaluate the function:

f(0) 0 f(0.5) 0.25 f(1) 1 f(1.5) 2.25 f(2) 4

Apply the trapezoidal rule:

text{Integral} approx frac{0.5}{2} left [0 2(0.25 1 2.25) 4 right] 1 cdot [0 23.5 4] 1 cdot [7 4] 1.1

Thus, the estimated integral is approximately 2.

This method provides a straightforward way to estimate integrals when exact solutions are complex or impractical.