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The Trapezoidal Rule is a numerical method used to estimate the value of a definite integral. This method approximates the area under a curve by dividing it into trapezoids rather than rectangles. It's an improvement over the basic rectangle method because it captures the area under linear segments of the curve more accurately.

Here's how it works:

• First, divide the interval \( [a, b] \) into \( n \) equal subintervals, each of width \( \Delta x = \frac{b-a}{n} \).
• Then, at each subinterval, compute the function values at the endpoints.
• The trapezoidal area for each subinterval is calculated and summed to estimate the total area.

The formula is:\[ T_n = \frac{b-a}{2n} \left( f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) \]In our example, using \( n = 2 \), we computed the trapezoidal approximation for \( \int_{1}^{3}(2x-1) \, dx \) and found it to be 8. This is close to the exact integral value, showing the method's efficiency.

Simpson's Rule offers another approach to numerical integration, providing more accuracy than the Trapezoidal Rule by incorporating parabolic segments instead of linear ones. This rule is particularly useful when the function is smooth and continuous.

The steps involve:

• First, ensure \( n \) is even for applying Simpson's Rule. Like before, the interval \( [a, b] \) is divided into \( n \) equal parts.
• Use the endpoints and midpoints of these intervals to fit a parabolic curve.
• Apply Simpson's Rule formula:\[ S_n = \frac{b-a}{3n} \left( f(a) + 4\sum_{i=1, \text{odd}}^{n-1} f(x_i) + 2\sum_{i=2, \text{even}}^{n-2} f(x_i) + f(b) \right) \]

For our example, with \( n = 2 \), the calculation again returned an approximation of 8 for \( \int_{1}^{3}(2x-1) \, dx \). This demonstrates the accuracy of Simpson's Rule even with a small number of intervals.

A definite integral, in mathematics, is used to calculate the total accumulation of a quantity, such as area under a curve, over a specific interval. It is represented as \( \int_{a}^{b} f(x) \, dx \), where \( [a, b] \) is the interval, and \( f(x) \) is the function being integrated.

The importance of definite integrals lies in their ability to:

• Provide the exact area under the curve or net change in whatever is being studied.
• Offer a foundation for understanding concepts such as accumulated distance, volume, and many physical phenomena.

In the context of the exercise, the definite integral \( \int_{1}^{3} (2x - 1) \, dx \) is evaluated to determine the precise area under the line \( y = 2x - 1 \) from \( x = 1 \) to \( x = 3 \). The calculation using antiderivatives yields an exact integral value of 6, verifying the approximate methods provided by the Trapezoidal and Simpson's Rule.