AP Calculus AB 6.2 Trapezoidal Rule to Approximate the Definite Integral

AP Calculus AB Topic 6.2 covers the Trapezoidal Rule as a method for approximating the value of a definite integral. Here’s a detailed overview of this topic:

### Trapezoidal Rule

1. **Concept**: The Trapezoidal Rule is a numerical integration technique that estimates the area under a curve by approximating it with trapezoids rather than rectangles. This method provides a more accurate approximation than Riemann sums, particularly for functions that are linear or nearly linear over small intervals.

2. **Procedure**:
- **Choose the Interval**: Start with a continuous function \( f(x) \) defined on a closed interval \([a, b]\).
- **Divide the Interval**: Split the interval into \( n \) equal subintervals, each of width \( \Delta x = \frac{b - a}{n} \).
- **Identify the Endpoints**: The endpoints of the subintervals are given by \( x_0 = a, x_1, x_2, \ldots, x_n = b \).
- **Calculate the Heights**: For each endpoint, calculate the function values \( f(x_0), f(x_1), \ldots, f(x_n) \).
- **Use the Trapezoidal Area Formula**: The area of each trapezoid can be calculated using the formula:
\[
\text{Area of trapezoid} = \frac{1}{2} \times (f(x_i) + f(x_{i+1})) \Delta x
\]
- **Sum the Areas**: The Trapezoidal Rule approximation is then calculated as:
\[
T_n = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)
\]
- This sum approximates the area under the curve from \( x = a \) to \( x = b \).

3. **Limit and Definite Integral**: As the number of subintervals \( n \) approaches infinity (and thus \( \Delta x \) approaches zero), the Trapezoidal Rule approximation approaches the exact value of the definite integral:
\[
\int_{a}^{b} f(x) \, dx \approx T_n
\]

4. **Applications**: The Trapezoidal Rule is particularly useful when the function is not easily integrable or when numerical methods are required for computation. It is widely used in various fields, such as physics and engineering, for estimating quantities like distance, area, and volume.

5. **Accuracy**: The accuracy of the Trapezoidal Rule generally improves with the number of subintervals \( n \). The error can be affected by the curvature of the function: if the function is linear, the trapezoidal approximation is exact; for non-linear functions, the approximation can still be quite good, especially with more subintervals.

### Conclusion
Understanding the Trapezoidal Rule provides students with a valuable technique for approximating definite integrals, enhancing their numerical integration skills. This method serves as a complement to Riemann sums and emphasizes the importance of estimating areas under curves in calculus, allowing students to apply these concepts to real-world problems.

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