# 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 `i`th 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.