Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{1}^{2} \frac{1}{x^{2}} d x, n=4 $$

Step-by-Step Solution

Answer

This solution requires calculations using the given formulas for the trapezoidal rule, Simpson's rule and the exact integral value. To give a short answer field based on this, the actual numerical values computed need to be provided. Hence, a meaningful short answer can't be provided right now.

Step 1: Analyze the Integral

Identify the function, which is \(f(x) = \frac{1}{x^{2}}\) and the interval of integration, which is from 1 to 2. The number of subdivisions, \(n\), is given as 4.

Step 2: Apply the Trapezoidal Rule

The formula for the Trapezoidal Rule is \(\frac{h}{2} \left[ f(a) + 2 \sum_{i=1}^{n-1} f \left( a + i \cdot h \right) + f(b) \right]\), where \(h = \frac{b-a}{n}\). In this case, \(a = 1, b = 2\), and \(n = 4\), so \(h = \frac{1}{4}\). Substituting these values into the formula and calculating, the approximation via the Trapezoidal Rule can be obtained.

Step 3: Apply Simpson's Rule

The formula for Simpson's Rule is \(\frac{h}{3} \left[ f(a) + 4 \sum_{i=1}^{n/2} f \left( a + (2i - 1) \cdot h \right) + 2 \sum_{i=1}^{n/2 - 1} f \left( a + 2i \cdot h \right) + f(b) \right]\), where \(h = \frac{b-a}{n}\). As before, \(a = 1, b = 2\), and \(n = 4\), so \(h = \frac{1}{4}\). Substituting these values into the formula and calculating, the approximation via Simpson's Rule can be obtained.

Step 4: Calculate the Exact Value of the Integral

The definite integral \(\int_{a}^{b} \frac{1}{x^{2}} dx\) equals \(- \left[ \frac{1}{x} \right]_{1}^{2}\). Evaluating this expression yields the exact value of the integral.

Step 5: Compare the Results

Compared the values obtained from the trapezoidal rule, Simpson's rule, and the exact value. Note down any differences and similarities, rounding all results to four decimal places.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a method in numerical analysis for approximating the value of a definite integral. It works by dividing the area under a curve into trapezoids, which are easier to calculate. Let's consider the function you're working with, which is \( f(x) = \frac{1}{x^{2}} \).

When applying the Trapezoidal Rule, you need to split the interval from 1 to 2 into smaller subintervals. In your exercise, the number of subdivisions, or trapezoids, is 4. Each trapezoid has a height \( h = \frac{1}{4} \) because the total interval length of 1 (from 1 to 2) is divided by 4. The area of each trapezoid is then calculated and summed to approximate the integral. Remember, the more trapezoids you use, the more accurate your approximation will be, up to a certain extent.

Simpson's Rule

Contrary to the Trapezoidal Rule, Simpson's Rule estimates the area under a curve by approximating the segments of the curve with parabolas instead of simple straight lines. This often leads to a more accurate approximation, especially when dealing with smooth functions.

Simpson's Rule requires an even number of intervals (in your case, 4), and it applies a weighted average to the heights of the parabolas formed over each subinterval. The formula involves both odd and even terms with multipliers of 4 and 2, respectively, except for the first and last terms. The intriguing aspect of Simpson's Rule is that by using parabolic arcs, it can provide a remarkably accurate approximation with fewer subintervals compared to the Trapezoidal Rule.

Definite Integral Approximation

Definite integral approximation is a cornerstone of calculus, particularly when you're dealing with functions that are difficult to integrate analytically or when you only have data points. Numerical integration methods like the Trapezoidal Rule and Simpson's Rule provide practical ways to approximate the integral's value.

The process involves summing up the areas of geometric shapes that closely fit under the curve being integrated over a specified interval. While these methods do not yield the exact integral value, they are highly useful for practical engineering and scientific calculations where an approximate value within an acceptable error margin is satisfactory.

Calculus

Calculus is the mathematical study of change, motion, and growth, and it is divided into two branches: differential calculus and integral calculus. Differential calculus concerns itself with the rate of change, while integral calculus deals with the accumulation of quantities.

The problem you're solving belongs to integral calculus, which focuses on finding the total size or value, such as areas under curves, volumes, and other accumulations. Both the Trapezoidal Rule and Simpson's Rule are rooted in integral calculus and showcase its application to real-world problems where exact solutions may be elusive or impractical to find, but approximate solutions are still incredibly valuable.