Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{3} \frac{x}{2+x+x^{2}} d x, n=6 $$

Step-by-Step Solution

Answer

Trapezoidal Rule results in an approximation of the integral of \(f(x)\) over the interval [0,3] and Simpson's Rule results in an approximation of the integral over the same interval, both rounded to three significant figures.

Step 1: Apply the Trapezoidal Rule

First, we calculate the width of each trapezoid: \(h = \frac{b-a}{n} = \frac{3-0}{6} = 0.5\). Then we apply the Trapezoidal Rule: \[T = \frac{h}{2}[(f(a) + f(b)) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5)]\] where \(x_i = a + ih\). The x-values are x_1 = 0.5, x_2 = 1, x_3 = 1.5, x_4 = 2, x_5 = 2.5. Substituting these values and evaluating the function at these points gives the approximation of the integral using the Trapezoidal Rule.

Step 2: Apply Simpson's Rule

Next, we use Simpson's Rule which implicates an arrangement of sections into pairs. The formula for Simpson's Rule is: \[S = \frac{h}{3}[(f(a) + f(b)) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5)]\] Again substitute these values and evaluate the function at these points to provide an approximation of the integral using Simpson's Rule.

Step 3: Round Solutions to Three Significant Figures

In our final step, both of these approximations must be rounded to three significant figures as instructed in the exercise.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a straightforward method for approximating the definite integral of a function. This rule involves dividing the area under the curve into a series of trapezoids, which makes it easier to sum up the areas. Here's how it works:

• First, determine the width of each trapezoid, which is called the step size or \( h \). It is found using the formula \( h = \frac{b-a}{n} \), where \( a \) is the lower limit, \( b \) is the upper limit, and \( n \) is the number of intervals.
• In the given example, \( h = 0.5 \), since the interval from 0 to 3 is divided into six parts.
• The Trapezoidal Rule formula is: \[T = \frac{h}{2}[(f(a) + f(b)) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1})]\]This formula accounts for the endpoints and twice the function values at the intermediate points.
• By calculating each term, you sum the area of each trapezoid to approximate the whole integral.

This method offers a good estimate, especially if the function is close to linear across intervals.

Simpson's Rule

Simpson's Rule is a more sophisticated method for numerical integration and provides greater accuracy than the Trapezoidal Rule. It uses parabolic segments instead of straight lines to approximate the curve. Here’s how Simpson's Rule works:

• Simpson’s Rule applies when \( n \) is even, dividing the interval into pairs to fit parabolas.
• The formula is: \[S = \frac{h}{3}[(f(a) + f(b)) + 4(f(x_1) + f(x_3) + \ldots) + 2(f(x_2) + f(x_4) + \ldots)]\]
• In this approach, alternate function evaluations are weighted more heavily. This characteristic helps accommodate more complex curves.
• In our example, each interval of 0.5 in width fits well into the Simpson’s Rule formulation, enhancing precision.

Simpson's Rule is excellent for functions that are polynomial in nature or when we desire more accuracy compared to other basic numerical methods.

Definite Integral

The concept of a definite integral is a cornerstone in calculus that represents the accumulation of quantities, such as areas under a curve. It is denoted as \( \int_{a}^{b} f(x) \, dx \), where:

• \( a \) is the lower limit of integration, and \( b \) is the upper limit of integration.
• The function \( f(x) \) is integrated over the interval \([a, b]\).

Definite integrals can be thought of as the net area between the x-axis and the curve from point \( a \) to point \( b \). They are essential for finding physical quantities like distance, area, and volume.

Using numerical methods such as the Trapezoidal Rule and Simpson's Rule allow us to estimate these integrals when an analytical solution is difficult or impossible to determine. These methods break the curve into manageable segments, making the accumulation process more feasible.
Overall, definite integrals are a key mathematical concept that have practical applications in science, engineering, and beyond.