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Simpson's Rule is a powerful technique used in numerical integration to approximate the definite integral of a function. It is particularly useful when we cannot easily find an explicit solution through direct calculations. The technique is based on the idea of approximating segments of the curve with a series of parabolic arcs drawn across subintervals of the range of integration.
Simpson's Rule is applied when the number of subintervals \( n \) is even.
This approximation formula is given by:\[\text{Approximation} = \frac{\Delta x}{3} \left[ f(x_0) + 4\sum_{\text{odd } i} f(x_i) + 2\sum_{\text{even } i} f(x_i) + f(x_n) \right]\]where:

• \( \Delta x = \frac{b-a}{n} \) – the width of each subinterval.
• \( x_i = a + i\Delta x \) – the endpoints and midpoints of subintervals.

The rule combines the areas under parabolas for alternating odd and even-indexed points and it typically offers very high accuracy while still remaining relatively simple to apply.

A definite integral symbolically represents the area under a curve described by a function over a specific interval \([a, b]\). It's written as:\[ \int_{a}^{b} f(x) \, dx \]where \( f(x) \) is the function being integrated. The main focus is to find the total accumulation of quantities, like area or volume.

• In comparison to an indefinite integral, which includes a constant of integration, a definite integral provides a numeric value.
• The limits of integration, \(a\) and \(b\), define the interval over which we perform the integration.

For instance, when computing \( \int_{0}^{2} (1+x^3) \, dx \), the limits specify that we are determining the total area from \(x = 0\) to \(x = 2\). Calculating this definite integral gives the exact value and helps verify numerical approximation methods like Simpson's Rule.

Direct integration is the classical method of finding an exact solution for a definite integral. It involves finding the antiderivative (or integral) of the function, and then evaluating this antiderivative at the upper and lower bounds of the interval.To directly integrate a function like \(1 + x^3\) over the interval [0, 2], we first find the antiderivative:\[ \int (1 + x^3) \, dx = x + \frac{x^4}{4} \]After finding the antiderivative, we apply the limits of integration using the Fundamental Theorem of Calculus:\[ \left.\left(x + \frac{x^4}{4} \right) \right|_0^2 \]Evaluating this expression at \(x = 2\) and \(x = 0\), we verify that the exact value of the integral is 6.
This technique not only affirms the accuracy of approximate methods like Simpson's Rule but also provides a clear and definitive result for the area under the curve.