The trapezoidal rule is a numerical method for approximating the definite integral of a function. It is particularly useful when the function is not easily integrable analytically. The trapezoidal rule approximates the area under a curve by dividing it into trapezoids rather than rectangles.

Definition of the Integral: We want to approximate the definite integral of a function \( f(x) \) over the interval \([a, b]\):

\[ I = \int_a^b f(x) \, dx \]

Divide the Interval: Divide the interval \([a, b]\) into \( n \) equal subintervals of width \( h \):

\[ h = \frac{b - a}{n} \]

The points dividing the intervals are:

\[ x_0 = a, x_1 = a + h, x_2 = a + 2h, \ldots, x_n = b \]

Trapezoidal Approximation: The area under the curve on each subinterval \([x_i, x_{i+1}]\) can be approximated by the area of the trapezoid formed by the line segments joining the points \((x_i, f(x_i))\) and \((x_{i+1}, f(x_{i+1}))\).

The area \( A_i \) of each trapezoid is given by:

\[ A_i = \frac{1}{2} \left( f(x_i) + f(x_{i+1}) \right) h \]

Sum the Areas: The total area \( I \) can be approximated by summing the areas of all trapezoids:

\[ I \approx \sum_{i=0}^{n-1} A_i = \sum_{i=0}^{n-1} \frac{1}{2} \left( f(x_i) + f(x_{i+1}) \right) h \]

Simplify the Summation: Factor out \( \frac{h}{2} \):

\[ I \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]

This leads to the final form of the trapezoidal rule:

\[ I \approx \frac{b - a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(a + ih) + f(b) \right) \]