Trapezoidal Rule | Numerical Methods Lesson

The trapezoidal rule is a numerical integration method that approximates the area under the curve of a function by dividing it into a number of trapezoids and then summing their areas. Here are the steps to apply it:

  1. Define the interval and partition: Select the interval [a, b] that you are interested in integrating over. Decide how many trapezoids you wish to use, n. The more trapezoids you use, the more accurate your approximation will be, but the more computationally expensive the process will become. Divide the interval into n equal subintervals. The width of each trapezoid, h, is then (b - a) / n.
  2. Evaluate the function at each point:
    For each of the n+1 points x_0, x_1, ..., x_n (where x_0 = a,
    x_i = a + i*h for i = 1, ..., n-1, and
    x_n = b), evaluate the function f(x).
    This gives you f(x_0), f(x_1), ..., f(x_n).
  3. Calculate the area of each trapezoid and sum them up: The area of each trapezoid can be calculated using the formula for the area of a trapezoid, (base1 + base2) / 2 * height. In this context, ‘base1’ and ‘base2’ are the function values at the two ends of the subinterval, and ‘height’ is the width of the subinterval h. So for each i = 0, ..., n-1, the area of the ith trapezoid is (f(x_i) + f(x_{i+1})) / 2 * h. Add up all these areas to get the total estimated integral of f over [a, b].

The above steps provide a simple method for approximating a definite integral using the trapezoidal rule. This method works best when the function being integrated is relatively smooth and does not change rapidly or unpredictably over the interval [a, b]. In other cases, more sophisticated methods like Simpson’s rule or Gaussian quadrature might be more appropriate.

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