91Ó°ÊÓ

The Trapezoidal Rule is a method of numerical integration, which is used to estimate the definite integral of a real-valued function. Its main idea is to approximate the region under the curve as a series of trapezoids, rather than using straightforward rectangles.
The general formula for the Trapezoidal Rule is:

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

This formula calculates the integral by breaking the interval \([a, b]\) into smaller sections with equal width. The width of each subinterval, \(h\), is determined by dividing the total interval by the number of trapezoids \(n\).

• Start by calculating \(h = \frac{b-a}{n} \).
• Next, evaluate the function at each of the division points \(x_0, x_1, \ldots, x_n \).
• Finally, plug these values into the Trapezoidal formula to get an approximate integral.

In our example, with \(f(x) = x^2\) over \([0, 3]\), it resulted in an estimate of \(10.96875\). The simple steps and the symmetry of the trapezoids make this method easy to comprehend yet quite effective.

Simpson's Rule is a more advanced approach to numerical integration compared to the Trapezoidal Rule. By fitting a second-order polynomial over the pairs of subintervals, it can achieve greater accuracy if the function is smooth.
For Simpson's Rule, the formula is:

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

This rule requires an even number of subintervals. It uses the points on the curve to form parabolic surfaces that better fit the function as opposed to the straight lines in the trapezoidal approach.

• Determine \(h = \frac{b-a}{n} \), as before.
• The function values at these points, especially focusing on every other point for weighting by 2 and 4, are inserted into Simpson's formula.
• This results in an approximation for the integral, which in our exercise is \(9.28125\).

Simpson's Rule generally provides a more accurate result, particularly when dealing with smooth curves, because the parabolas more closely follow the function's shape.

Error Bound Computation is a crucial part of numerical integration, as it offers a way to estimate the possible deviation from the true integral value.
For the Trapezoidal Rule, the error bound is calculated as:

• \( |E_T| \leq \frac{(b-a)^3}{12n^2} |f''(x)|_{max} \)

This computation uses the second derivative of the function \(f(x)\) because it gives us an idea of how much the function is curved. For a function \(f(x) = x^2\), \(f''(x) = 2\), which results in an error bound of approximately 0.140625 for our problem.
When using Simpson's Rule, the error bound changes to:

• \( |E_S| \leq \frac{(b-a)^5}{180n^4} |f^{(4)}(x)|_{max} \)

This involves the fourth derivative, which often is zero for polynomial functions of degree less than four, leading to no theoretical error as in our case. In practical applications, knowing the error bound helps users understand how reliable their numerical result is; for Simpson's Rule, when the fourth derivative is zero, it implies the approximation is very close to the actual integral.