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:

**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`

.**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)`

.**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.

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