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The Trapezoidal Rule is a numerical integration technique used to approximate the definite integral of a function. It achieves this by approximating the region under the curve as a series of connected trapezoids, rather than rectangles, which can sometimes provide more accuracy. To use the Trapezoidal Rule, one divides the interval \([a, b]\) into \(n\) equal subintervals of width \(\Delta x = \frac{b-a}{n}\).
The approximation is calculated using the formula:

• \( T_n = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \)

If the function is increasing or decreasing, the trapezoids can overestimate or underestimate the area. The rule becomes more accurate as \(n\) increases, which reduces subinterval width \(\Delta x\), resulting in a closer approximation of the true curve.
In this exercise, doubling \(n\) reduces the error, making the approximation with \(T_{12}\) more accurate than \(T_6\).

The Midpoint Rule is another method for approximating the integral of a function, using rectangles where the height of each rectangle is determined by the value of the function at the midpoint of each interval.
The process involves dividing the given interval \([a, b]\) into \(n\) equal parts, similar to other methods, and is characterized by the formula:

• \( M_n = \Delta x \sum_{i=1}^{n} f(x_i^*) \)

where \(x_i^*\) represents each midpoint in the interval. This method is often more efficient when the function is reasonably smooth as it centers each interval segment on the curve, negotiating the variations better than endpoints would.
With increased \(n\), the Midpoint Rule's precision improves significantly, as smaller intervals mean the midpoints give a better representation of the curve. In the exercise, moving from \(M_6\) to \(M_{12}\) shows a noticeable decrease in the error.

Simpson’s Rule provides one of the most accurate approaches among these numerical integration techniques, especially when dealing with polynomial functions. It approximates the function by fitting a quadratic polynomial through three consecutive points and then calculating the area under this parabolic segment. \(n\) should be even for Simpson's Rule, as it operates over pairs of intervals.
The general formula is:

• \( S_n = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{\text{odd}} f(x_i) + 2 \sum_{\text{even}} f(x_i) + f(x_n) \right) \)

This formula implies that different weights are applied to the function's value at odd and even intervals, making it superior in handling curvatures of graphs efficiently.

Simpson's Rule minimizes the error in approximation greatly as \(n\) increases. When the number of subintervals doubles, for example from \(n=6\) to \(n=12\), the error generally reduces substantially, confirming its strength in achieving higher accuracy with fewer calculations than other rules like Trapezoidal or Midpoint.