91Ó°ÊÓ

Numerical integration is a handy way of approximating the area under a curve. The Trapezoidal Rule is one such method that divides the area into trapezoids rather than rectangles. For a function, say \( f(x) = x^4 \), between limits 0 and 2, the Trapezoidal Rule estimates the integral by creating trapezoids under the curve, summing their areas.

The formula for this approximation is:\[T_n = \frac{b-a}{2n} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)\]where \( n \) is the number of intervals, \( b \) and \( a \) are the upper and lower limits, respectively, and \( x_i \) are the dividing points.

Key points to remember:

• Increases in \( n \) (number of intervals) make the approximation closer to the true value.
• The error is generally proportional to \( \frac{1}{n^2} \). So, doubling \( n \) often decreases the error significantly, but not as much as Simpson's rule.

Using this rule for \( n=6 \) and \( n=12 \) shows how dividing the function into more intervals provides a better approximation, bringing \( T_n \) closer to the exact value of 6.4.

The Midpoint Rule is another method of numerical integration that refines the approximation of definite integrals. Unlike the Trapezoidal Rule, which uses endpoints of intervals, the Midpoint Rule uses the midpoint of each subinterval to approximate the area under the curve. This method can be particularly effective when dealing with smooth functions like \( f(x) = x^4 \).

Here's the formula:\[M_n = \frac{b-a}{n} \sum_{i=0}^{n-1} f\left( \frac{x_i + x_{i+1}}{2} \right)\]Each term in the formula is the function value at the midpoint \( \frac{x_i + x_{i+1}}{2} \) multiplied by the width of the interval.

Observations to note:

• This approach can be a bit more accurate than the Trapezoidal Rule for certain functions, like polynomial functions.
• The error often decreases in proportion to \( \frac{1}{n^2} \), similar to the Trapezoidal Rule.

By increasing \( n \) from 6 to 12 in our example, you can see an improvement in the accuracy, as \( M_n \) approaches the exact integral value 6.4.

Simpson's Rule is an advanced technique in numerical integration, which offers remarkable accuracy by using parabolic arcs instead of straight lines to approximate the area under a curve. This rule is especially effective for the polynomial function \( f(x) = x^4 \) in our example.

The formula for Simpson's Rule is:\[S_n = \frac{b-a}{3n} \left( f(x_0) + 4 \sum_{i=1, \text{odd}}^{n-1} f(x_i) + 2 \sum_{i=2, \text{even}}^{n-2} f(x_i) + f(x_n) \right)\]What makes Simpson's Rule stand out is its error rate, which diminishes proportionally to \( \frac{1}{n^4} \). This makes it substantially more precise, especially when \( n \) is increased.

Keep in mind:

• Simpson's Rule typically provides a superior approximation, primarily due to its higher degree of precision for polynomial integrands.
• The method is efficient for even values of \( n \).

Applying this rule as \( n \) doubles from 6 to 12 reveals a robust reduction in error, making \( S_n \) practically equal to the exact value of 6.4, showcasing why it's favored for meticulous calculations.