Problem 5 - 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. $$ \int_{0}^{2} x^{3} d x, n=8 $$

Step-by-Step Solution

Answer

Your results will depend on the precise calculations. You should find that both the Trapezoidal Rule and Simpson's Rule provide reasonably good approximations to the exact value of the integral, however, Simpson's Rule might be more accurate due to the nature of quadratic interpolating polynomial, which gives a better approximation for smooth or continuous functions.

Step 1: Setup

We first set up the integral, identify the upper limit \(b=2\), lower limit \(a=0\), and the number of intervals \(n=8\). Then, compute the width of intervals \(h=(b-a)/n\) for both methods.

Step 2: Application of the Trapezoidal Rule

The general formula for the trapezoidal rule is \(\frac{h}{2}[f(a)+2\sum_{i=1}^{n-1} f(a+ih)+f(b)]\). Substituting our function \(f(x) = x^{3}\), \(a\), \(b\), \(h\), and \(n\) into this formula we can calculate the approximation.

Step 3: Application of Simpson's Rule

Now switch to Simpson's Rule: \(\frac{h}{3}[f(a)+4\sum_{i=1,3,...,n-1} f(a+ih)+2\sum_{i=2,4,...,n-2} f(a+ih)+f(b)]\). Substitute \(f(x) = x^{3}\), \(a\), \(b\), \(h\), and \(n\) into this formula to get the approximation.

Step 4: Calculate Exact Value

Now, find the exact value of the definite integral by anti-differentiating \(x^{3}\), apply the limits of \(0\) to \(2\) to get the exact value.

Step 5: Comparison of Results

Finally, compare the approximations obtained from the Trapezoidal rule and Simpson's rule with the exact value from step 4. Notice the difference (error) between the numerical approximations and the exact value. This gives an indication of the level of accuracy of the numerical methods.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical technique used to approximate the value of definite integrals. It's especially handy when finding the exact integral analytically is complex or impossible. The basic idea is to divide the area under the curve into smaller trapezoids rather than rectangles, which gives a better approximation of the area.

Here’s how it works:

• First, determine the width of each interval, which is \( h = \frac{b-a}{n} \). Here, \(a\) and \(b\) are the lower and upper limits of the integral, and \(n\) is the number of subintervals.
• The function to be integrated is evaluated at each interval point \( f(a), f(a+h), f(a+2h), \ldots, f(b) \).
• Apply the formula: \[\frac{h}{2}[f(a) + 2f(a+h) + 2f(a+2h) + \ldots + 2f(b-h) + f(b)]\]

This method gives a linear approximation between each pair of points, reducing the error compared to simple rectangles. The more intervals used (higher \(n\)), the better the approximation.

Simpson's Rule

Simpson's Rule offers an even more accurate way to approximate definite integrals. It improves upon the Trapezoidal Rule by using parabolic segments rather than straight lines to approximate the curve. This rule works best when the number of intervals \(n\) is even.

Here’s the procedure:

• Calculate the width of intervals just like in the Trapezoidal rule: \( h = \frac{b-a}{n} \).
• Evaluate the function at each interval point, but with a twist. You'll use your results differently:
  • Start and end points: \( f(a) \) and \( f(b) \).
  • Odd intervals: Multiply by 4. These are the points where the subinterval index is odd \((a+h, a+3h, \ldots)\).
  • Even intervals: Multiply by 2. These are the points where the index is even \((a+2h, a+4h, \ldots)\).

The formula becomes: \[\frac{h}{3}[f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + \ldots + 4f(b-h) + f(b)]\]This approach typically provides a more precise approximation, especially for smooth functions.

Definite Integral

A definite integral computes the area under a curve between two points, providing a numerical value. It’s a core concept in calculus, used widely across various fields such as physics, engineering, and statistics.

Here’s what you need to know:

• It’s denoted as \( \int_{a}^{b} f(x) \, dx \), where \(a\) is the lower limit and \(b\) is the upper limit.
• The process of integration involves finding the antiderivative of the function, which is then evaluated at the limits.
• The exact value can be compared with numerical approximations, providing insight into how close approximations like those from the Trapezoidal and Simpson’s Rule can be to the actual value.
• For functions that are easy to integrate, the definite integral provides the exact answer right away without approximations.

In practice, when a function is complex, numerical methods, like the ones discussed, become practical tools to estimate the integral.