Problem 46 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Determine how many subintervals are required to guarantee accuracy to within \(0.00005\) of the actual value of the given integral using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{0.5}^{1} e^{-1.1 x} d x $$

Step-by-Step Solution

Answer

Put your short answer here. Short answer must be a very short answer to the question based on the solution steps

Step 1: Understand the formulas

To determine the number of subintervals for the trapezoidal and Simpson's rule, we use the error bounds formulas for each method.

Step 2: Error bound for the trapezoidal rule

The error bound for the trapezoidal rule is given by:

Step 3 Title

Content for Step 3

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical method to approximate the value of a definite integral. Think of it as breaking down the area under a curve into small trapezoids and summing their areas to estimate the total integral. This method is simple and effective for many functions.

To apply the trapezoidal rule, we divide the interval \([a, b]\) into \(n\) subintervals of equal width \(\frac{b-a}{n} \). Each subinterval has a height that corresponds to the function value at the subinterval endpoints. The formula is given by:

\[ T_n = \frac{b-a}{2n} \bigg[ f(a) + 2 \sum_{i=1}^{n-1} f(a+i \Delta x) + f(b) \bigg] \]

Where \( \Delta x = \frac{b-a}{n} \) is the width of each subinterval.

In the context of the original exercise, to determine how many subintervals are required to achieve a certain accuracy, we use the error bound for the trapezoidal rule: \[ E_T \leq \frac{(b-a)^3}{12n^2}f''(\theta) \]

Here, \( \theta \) is some point in the interval \([a, b]\), and \((b-a)\) is the length of the interval.

Simpson's Rule

Simpson's rule is another method to approximate the value of a definite integral. It provides a more accurate approximation compared to the trapezoidal rule by using parabolic segments instead of straight lines.

To apply Simpson's rule, we also divide the interval \([a, b]\) into an even number of subintervals (n). The formula for Simpson's rule is:

\[ S_n = \frac{b-a}{3n} \bigg[ f(a) + 4 \sum_{i=1, \text{odd}}^{n-1} f(a+i \Delta x) + 2 \sum_{i=2, \text{even}}^{n-2} f(a+i \Delta x) + f(b) \bigg] \]

In this method, the subintervals alternate between coefficients of 4 and 2. This distribution helps us capture more area under the curve, leading to a more accurate approximation.

To achieve a desired accuracy, we use the error bound for Simpson's rule:

\[ E_S \leq \frac{(b-a)^5}{180n^4}f^{(4)}(\theta) \]

Here, \( f^{(4)}(\theta) \) represents the fourth derivative of the function \( f \) evaluated at some point \(\theta \) in the interval \([a, b]\).

Error Bound

The error bound formulas provide an estimate of the maximum possible error when using numerical integration methods like the trapezoidal rule and Simpson's rule. These formulas depend on derivatives of the function being integrated.

The error bound for the trapezoidal rule is calculated using the second derivative of the function \( f \). This gives the maximum error in approximating the integral. For the original exercise, if we want this error to be less than \( 0.00005 \), we solve:

\[ \frac{(b-a)^3}{12n^2} \times \text{max}(|f''(\theta)|) \leq 0.00005 \]

For Simpson's rule, the error bound involves the fourth derivative of the function. We ensure the maximum error is within the desired accuracy by solving:

\[ \frac{(b-a)^5}{180n^4} \times \text{max}(|f^{(4)}(\theta)|) \leq 0.00005 \]

Using these bounds, we determine the required number of subintervals \( n \) to guarantee the approximation error stays within acceptable limits. This step is crucial for achieving precise results in numerical integration.