Problem 24 - Thomas Calculus

Question

In Exercises \(15-26,\) estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by ( a ) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(15-22\) are the integrals from Exercises \(1-8 .\) ) $$ \int_{0}^{3} \frac{1}{\sqrt{x+1}} d x $$

Step-by-Step Solution

Answer

For the Trapezoidal Rule, at least 712 subintervals are needed. For Simpson's Rule, at least 27 subintervals are needed.

Step 1: Determine the Error Formula for Trapezoidal Rule

The error bound for the Trapezoidal Rule is given by the formula: \[ E_T = \frac{(b-a)^3}{12n^2} \cdot K_2 \]where \( K_2 \) is the maximum value of the second derivative of the function \( f(x) \) on the interval \([a, b]\). For this problem, \( f(x) = \frac{1}{\sqrt{x+1}} \), \( a = 0 \), and \( b = 3 \). We need to find \( f''(x) \) and determine \( K_2 \).

Step 2: Calculate the Second Derivative

First, find \( f'(x) \):\[ f'(x) = \frac{d}{dx}\left(x+1\right)^{-1/2} = -\frac{1}{2}(x+1)^{-3/2} \]Now find \( f''(x) \):\[ f''(x) = \frac{d}{dx}\left(-\frac{1}{2}(x+1)^{-3/2}\right) = \frac{3}{4}(x+1)^{-5/2} \]We need to find the maximum of \( f''(x) \) on \([0, 3]\). Since \( (x+1)^{-5/2} \) decreases as \( x \) increases, \( K_2 = f''(0) = \frac{3}{4} \).

Step 3: Apply the Trapezoidal Rule Error Formula

Plug \( K_2 = \frac{3}{4} \), \( a = 0 \), and \( b = 3 \) into the error formula:\[ E_T = \frac{(3-0)^3}{12n^2} \cdot \frac{3}{4} \]We want \( E_T < 10^{-4} \), so:\[ \frac{27}{12n^2} \cdot \frac{3}{4} < 10^{-4} \]Simplify and solve for \( n \):\[ \frac{81}{16n^2} < 10^{-4} \]\[ n^2 > \frac{81 \times 10^4}{16} \]\[ n^2 > 506250 \]\[ n > \sqrt{506250} \approx 711.52 \]Thus, \( n \) must be at least 712.

Step 4: Determine the Error Formula for Simpson's Rule

The error bound for Simpson's Rule is given by the formula: \[ E_S = \frac{(b-a)^5}{180n^4} \cdot K_4 \]where \( K_4 \) is the maximum value of the fourth derivative of the function \( f(x) \) on the interval \([a, b]\). We now need to compute \( f^{(4)}(x) \) and determine \( K_4 \).

Step 5: Calculate the Fourth Derivative

Continue differentiating to find the fourth derivative. Previously, we found:\[ f''(x) = \frac{3}{4}(x+1)^{-5/2} \]Now find further derivatives:\[ f'''(x) = -\frac{15}{8}(x+1)^{-7/2}, \]\[ f^{(4)}(x) = \frac{105}{16}(x+1)^{-9/2} \]Since \( (x+1)^{-9/2} \) decreases as \( x \) increases, \( K_4 = f^{(4)}(0) = \frac{105}{16} \).

Step 6: Apply the Simpson's Rule Error Formula

Using \( K_4 = \frac{105}{16} \), \( a = 0 \), and \( b = 3 \), plug into the error formula:\[ E_S = \frac{(3-0)^5}{180n^4} \cdot \frac{105}{16} \]We want \( E_S < 10^{-4} \), so:\[ \frac{243 \times 105}{2880n^4} < 10^{-4} \]Simplify and solve for \( n \):\[ n^4 > \frac{243 \times 105 \times 10^4}{2880} \]\[ n^4 > 886718.8 \]\[ n > \sqrt[4]{886718.8} \approx 26.1 \]Thus, \( n \) must be at least 27.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a technique for estimating definite integrals. It works by approximating the region under a curve as a series of trapezoids, making it easier to calculate. This method comes in handy when we are looking to approximate the area under a curve when it’s difficult or impossible to find the exact integral.

To apply the Trapezoidal Rule, you need to divide the region you want to integrate into a certain number of subintervals, known as \( n \). You then calculate the heights of the trapezoids at each point by using the value of the function at those points.

• The more subintervals you use, the more accurate the approximation.
• Each subinterval is used to form one trapezoid.
• To reduce error, the subintervals should be evenly spaced.

The error bound for this method is calculated with the formula \( E_T = \frac{(b-a)^3}{12n^2} \cdot K_2 \), where \( K_2 \) is the maximum of the second derivative of the function on the interval \[a, b\].

In the exercise, we determined that the number of trapezoids required was at least 712 to ensure the error is less than \(10^{-4}\). More trapezoids, meaning a larger \( n \), results in a smaller error and a more accurate approximation.

Simpson's Rule

Simpson’s Rule is another method to approximate integrals, similar to the Trapezoidal Rule but often more accurate with fewer evaluations. Instead of forming trapezoids, it uses parabolas to approximate the area under the curve, matching the function at three points in each subinterval.

This method requires an even number of subintervals, also known as \( n \). By using parabolas, Simpson's Rule can provide a more refined approximation compared to simple linear trapezoids.

• The rule is particularly effective for functions that are smooth and well-behaved over the region of integration.
• The formula for the error in Simpson’s Rule is \( E_S = \frac{(b-a)^5}{180n^4} \cdot K_4 \).
• \( K_4 \) is the maximum of the fourth derivative of your function on the interval \([a, b]\).

By calculating \( K_4 \) and applying it to the formula, the exercise reveals that we need at least 27 subintervals to keep the error under \(10^{-4}\). This demonstrates the efficiency of Simpson's Rule in achieving low error rates with fewer subintervals.

Error Estimation

Error estimation is crucial in numerical integration as it helps you determine how close your approximation is to the actual value. Both the Trapezoidal Rule and Simpson’s Rule provide specific formulas for estimating the error margin, which depends on derivatives of the function being integrated.

When using these rules, the error is influenced by multiple factors:

• The function's behavior - smoother functions tend to give smaller errors.
• Order of derivatives used in error formulas - higher derivatives often implicate faster error convergence.
• Number of subintervals \( n \) - increasing \( n \) typically reduces the error.

Error formulas are derived from the Taylor series expansion and provide a way to predict how the approximation might deviate from the true value.

In the exercise, understanding how to apply these error estimation formulas guided the choice of subintervals, ensuring the resulting approximation met the error criteria. This showcases the importance of correctly estimating the error for reliable results.

Integral Approximation

Integral approximation is the process of estimating the value of a definite integral when it’s difficult to compute exactly. This approach becomes essential when dealing with complex or computationally expensive functions.

Several techniques exist, with the Trapezoidal Rule and Simpson’s Rule being popular due to their simplicity and efficiency. Approximation methods are invaluable:

• When the integrand is difficult to integrate analytically.
• For functions defined only at discrete points, making exact integration impossible.
• In applications where speed is more critical than precision.

The exercise demonstrated how to approximate \( \int_{0}^{3} \frac{1}{\sqrt{x+1}} \, dx \) by determining the appropriate number of subintervals to meet a specific error criterion. This illustrates how integral approximation techniques are applied in real-world scenarios where precise integration is not feasible.