91Ó°ÊÓ

The Midpoint Rule is a straightforward numerical integration technique used to estimate the area under a curve. This method involves dividing the interval of integration into smaller subintervals called partitions.
In our example, the interval \([-1, 1]\) is divided into \( n = 10 \) equal parts. The width of each subinterval, \( h \), is \( 0.2 \).
At the heart of this technique is the selection of midpoints from each subinterval. These midpoints, denoted as \( x_i^* \), are calculated as \( x_i^* = -1 + \left( i + \frac{1}{2} \right)h \).
The function value at each midpoint is then calculated using \( f(x_i^*) \), resulting in values like \[0.3704, 0.3846, \ldots, 0.5556\].
The formula \( M_{10} = h \sum_{i=0}^{9} f(x_i^*) \) helps in adding up all these contributions to get an estimate of \( 0.8047 \).
The Midpoint Rule typically gives a better approximation compared to some of the other elementary methods, especially when the function is smooth. In this specific exercise, it's interesting to note that the absolute error in using the Midpoint Rule was exactly \( 0.0000 \), an indication of its good precision for this integral.

The Trapezoidal Rule is another technique for numerical integration which approximates the area under a curve by summing up trapezoids rather than rectangles. Imagine slicing the area and sitting a bunch of trapezoids side by side. Each trapezoid approximates the function in a linear way.
In this exercise, like the Midpoint Rule, we again divide the interval \([-1, 1]\) into \( n=10 \) partitions, with each subinterval having a width \( h = 0.2 \).
However, the key difference is that this rule averages the values of the function at the endpoints of each subinterval, calculated as \( x_i = -1 + 0.2i \), leading to the formula:
\( T_{10} = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{9} f(x_i) + f(x_{10}) \right) \).
This extends its calculations, like \[0.3333, 0.7407, \ldots, 1.4545\].
The approximation results in \( T_{10} \approx 0.8063 \).
A distinct attribute of the Trapezoidal Rule is that it tends to be more accurate when dealing with functions that are linear or nearly linear over the subintervals, but can sometimes overestimate for curvier functions. The observed absolute error here was \( 0.0016 \), hinting this is generally a decent approach but less precise than the Midpoint Rule for this function.

Simpson's Rule offers a more advanced approach to numerical integration by combining the Midpoint and Trapezoidal methods for greater accuracy.
This rule, in essence, uses parabolic arcs instead of straight lines to approximate the function over each subinterval.
The exercise requires the use of \( n = 10 \) subintervals, which is ideal because Simpson’s Rule works well when the number of subintervals is even.
Simpson's Rule formula is \( S_{10} = \frac{h}{3} \left( f(x_0) + 4 \sum_{i=odd} f(x_i) + 2 \sum_{i=even, i eq n} f(x_i) + f(x_n) \right) \), where \( h = 0.2 \) is once again used.
Applying this rule to our function \( f(x) = \frac{1}{2x+3} \), we compute values \[0.3333, 0.7407, \ldots, 1.4545\], multiplying by coefficients 4 and 2 as needed.
Simpson's Rule gives an approximation of \( S_{10} \approx 0.8047 \).
This method is particularly beneficial for functions that are quadratic or smooth—and here it showed no significant error \(|E_S| = 0.0000 \).
Generally, Simpson’s Rule is the preferred method for many integrals due to its higher degree of accuracy with a relatively simple formula.