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The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It works by approximating the area under the curve of a function by dividing it into trapezoids and summing up their areas.

The basic idea behind the trapezoidal rule is to approximate the curve with straight line segments connecting consecutive points on the curve. By considering each pair of adjacent points as the endpoints of a trapezoid, the area of each trapezoid can be calculated using the formula:

Area = (b - a) * (f(a) + f(b)) / 2

where 'a' and 'b' are the limits of integration, and f(a) and f(b) are the function values at those points.

To approximate the definite integral using the trapezoidal rule, you need to divide the interval [a, b] into smaller subintervals of equal width, typically denoted by 'h'. The number of subintervals is given by:

n = (b - a) / h

where 'n' represents the number of trapezoids. Then, the integral is approximated by summing up the areas of all the trapezoids:

Integral ≈ h * [(f(a) + f(a+h)) / 2 + (f(a+h) + f(a+2h)) / 2 + ... + (f(a+(n-1)h) + f(a+nh)) / 2]

In this formula, the term (f(a) + f(a+h)) / 2 represents the average of the function values at the two endpoints of the first trapezoid, (f(a+h) + f(a+2h)) / 2 represents the average of the function values at the endpoints of the second trapezoid, and so on.

As the number of subintervals 'n' increases (or equivalently, as the width of each subinterval 'h' decreases), the approximation of the integral becomes more accurate.

It's worth noting that the trapezoidal rule is a first-order approximation method, meaning its error decreases with the square of the step size. Therefore, if you want a more accurate result, you may need to use a smaller step size or consider using more advanced numerical integration techniques.