The Trapezoidal Rule and Simpson's Rule are numerical methods for approximating the definite integral of a function, particularly useful when dealing with complex or irregular functions that are difficult to integrate analytically.

1. Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by dividing the interval into smaller subintervals and approximating each part as a trapezoid, rather than a rectangle.

Given an integral of a function $f(x)$ over the interval $[a, b]$, $\int_a^b f(x) \, dx,$ we divide the interval $[a, b]$ into $n$ equal subintervals, each of width $h = \frac{b - a}{n}$.

The formula for the Trapezoidal Rule is: $\int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right),$ where $x_0 = a$, $x_n = b$, and $x_i = a + i \cdot h$.

2. Simpson's Rule

Simpson's Rule is a more accurate approximation method than the Trapezoidal Rule, especially for functions that are well approximated by parabolic curves over small intervals. It approximates the function using quadratic (parabolic) sections over each pair of subintervals.

For Simpson's Rule, the interval $[a, b]$ is divided into an even number of subintervals, with width $h = \frac{b - a}{n}$ where $n$ must be even.

The formula for Simpson's Rule is: $\int_a^b f(x) \, dx \approx \frac{h}{3} \left( f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right),$ where $x_0 = a$, $x_n = b$, and $x_i = a + i \cdot h$ as before.

Key Differences

1. Accuracy: Simpson's Rule generally gives a more accurate approximation than the Trapezoidal Rule, especially if the function is smooth.
2. Interval Requirements: Simpson’s Rule requires an even number of intervals, whereas the Trapezoidal Rule has no such requirement.
3. Approximation Shape: The Trapezoidal Rule uses straight lines (trapezoids), while Simpson’s Rule uses parabolic curves.

Would you like a detailed example for either rule, or do you have any questions?

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Here are some related questions to explore these rules further:

1. How does increasing the number of intervals $n$ affect the accuracy of the Trapezoidal and Simpson's Rule?
2. What are some common applications of the Trapezoidal Rule and Simpson's Rule in real-world scenarios?
3. Can we combine both methods for more accurate results, and if so, how?
4. How do we estimate the error for both the Trapezoidal Rule and Simpson’s Rule?
5. What adjustments are necessary if the function $f(x)$ is not smooth?

Tip: Both rules can be improved by increasing the number of intervals, as this reduces the approximation error.