Problem 4 - 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. $$ \int_{1}^{2} \frac{1}{x} d x, n=4 $$

Step-by-Step Solution

Answer

After calculating, we get specific values for the integral using the Trapezoidal Rule, Simpson's Rule and the exact value. Comparison between these allows us to see the differences between the exact integral and the approximations provided by the numerical methods.

Step 1: Define the function and intervals

The function is \(f(x) = \frac{1}{x}\) and the interval for the definite integral is \([1, 2]\) with four divisions, so each sub-interval, \(\Delta x = \frac{b - a}{n} = \frac{2 - 1}{4} = 0.25\). Each of the four x-values then becomes \(x_0 = 1\), \(x_1 = 1.25\), \(x_2 = 1.5\), \(x_3 = 1.75\), and \(x_4 = 2\).

Step 2: Calculate using the Trapezoidal Rule

The formula for the Trapezoidal Rule is \(\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)]\). Substituting the known values gives us the result.

Step 3: Calculate using Simpson's Rule

The Simpson's Rule for \(n=4\) is \(\frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\). Substituting the known values gives us the result.

Step 4: Calculate the exact value of the definite integral

The exact value of the definite integral can be found by directly integrating the function \(f(x) = \frac{1}{x}\) between the limits 1 and 2.

Step 5: Compare the results

Now that we have estimated values using Trapezoidal and Simpson's Rules and the exact value of the definite integral, we can compare these results, all rounded to four decimal places

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method for approximating the value of a definite integral. It works by dividing the area under the curve into trapezoids, which are easier to calculate than curved shapes. Each trapezoid is formed by:

• Choosing points that partition the interval on the x-axis.
• Calculating the corresponding y-values of the function.
• Applying the formula to find the area.

The formula for the Trapezoidal Rule when you have an interval \[a, b\]\, divided into \ n \ subintervals is:\[ T = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] \]
Here, \ \Delta x \ is the width of each subinterval. For the given function \ f(x) = \frac{1}{x} \, each value of \ f(x) \ at \ x_0, x_1, x_2, \ and so on, represents the height at that point.
By summing these trapezoids, you approximate the area under the curve, which gives an estimate of the integral. It's effective for smooth functions but can be less accurate for complex curves.

Simpson's Rule

Simpson's Rule is another approach for estimating definite integrals. This rule tends to be more accurate than the Trapezoidal Rule because it approximates the function by a series of parabolas rather than straight lines.
With Simpson's Rule, you use an even number of subintervals, \ n \, and the formula is:\[ S = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)] \]Each consecutive set of three points is used to form a parabola. This method captures the curve more accurately, especially when the function itself is quadratic or smooth.

• Provides better accuracy with fewer subintervals.
• Useful for functions that have continuous second derivatives.

In this exercise with \ f(x) = \frac{1}{x} \, applying Simpson's Rule helps achieve a closer approximation to the true integral value.

Definite Integral

A definite integral calculates the net area under a curve from one point to another on the x-axis. The limits of integration, \ a \ and \ b \, define the interval over which you're finding this area.
For the function \ f(x) = \frac{1}{x} \, the definite integral is expressed as:\[ \int_{1}^{2} \frac{1}{x} \, dx \]
This integral represents the area under the curve \ f(x) \ from \ x = 1 \ to \ x = 2 \, which can be found using methods like the Trapezoidal Rule, Simpson's Rule, or exact calculus.

• Helps in finding the total accumulation of a quantity.
• Can be interpreted geometrically as area or physically as total change over an interval.
• Essential part of calculus, linking derivatives to the total accumulation.

The exact value of this integral gives you the precise area, which can be compared to numerical approximations.