Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{1}^{3} \frac{1}{x} d x $$

Step-by-Step Solution

Answer

For the Trapezoidal Rule, the smallest \(n\) that ensures an error less than 0.0001 is the ceiling value of the square root of ((2*8)/(12*0.0001)). For the Simpson's Rule, the smallest \(n\) is the ceiling value of the fourth root of ((2*32)/(180*0.0001)). The calculations will yield actual numerical values for \(n\).

Step 1: Recall the Error Formulas

For the Trapezoidal Rule, the error formula is \(E_t = -\frac{(b-a)^3}{12n^2}f''(c)\), and for Simpson's Rule, it is \(E_s = -\frac{(b-a)^5}{180n^4}f^{(4)}(c)\), where \(a\) and \(b\) are the lower and upper limits of the integral, \(f''(c)\) and \(f^{(4)}(c)\) are the second and fourth derivative of \(f(x)\) evaluated at \(c\) within [a,b], and \(n\) is the number of subintervals

Step 2: Calculate Derivatives

Given \(f(x) = 1/x\), it's second derivative \(f''(x)\) is \(2/x^3\) and the fourth derivative \(f^{(4)}(x)\) is \(-24/x^5\). Evaluate these at \(c\), assuming the error reaching maximum at the endpoint, \(c=a=1\). Then, \(f''(1) = 2\) and \(f^{(4)}(1) = -24\)

Step 3: Use the Error Formulas for Trapezoidal Rule

Substitute \(a=1, b=3, f''(1) = 2\) into Trapezoidal Rule error formula, and set it less than 0.0001. \(|-((3-1)^3/12n^2)*2| < 0.0001\). Solve for \(n\), the result is \(n > \sqrt{(2*2^3)/(12*0.0001)}\). So the smallest integer \(n\) that makes error less than 0.0001 is the ceiling of this value.

Step 4: Use the Error Formulas for Simpson's Rule

Substitute \(a=1, b=3, f^{(4)}(1) = -24\) into Simpson's Rule error formula, and set it less than 0.0001. \(|-((3-1)^5/180n^4)*(-24)| < 0.0001\). Solve for \(n\), the result is \(n > \sqrt[4]{((2*2^5)/180*0.0001)}\). So the smallest integer \(n\) that makes error less than 0.0001 is the ceiling of this value.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a technique for approximating the definite integral of a function, which involves dividing the area under the curve into a series of trapezoids. By calculating the area of each trapezoid, we can approximate the total area under the curve. This method is especially useful when dealing with numerical integration applications in real-world scenarios.
The general formula for the Trapezoidal Rule is \[ T_n = \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right), \]where \( n \) is the number of trapezoids, \( a \) and \( b \) are the limits of integration, and \( f(x_i) \) represents the function's value at each point \( x_i \).
To measure the accuracy of this approximation, we use an error formula:\[ E_t = -\frac{(b-a)^3}{12n^2}f''(c), \]where \( f''(c) \) is the second derivative of the function at some point \( c \) within the interval \([a,b]\). The key to reducing the error is to increase \( n \), thus using more trapezoids which results in a more accurate approximation.

Simpson's Rule

Simpson's Rule is another method for numerical integration, which often provides a higher accuracy than the Trapezoidal Rule. It works by approximating the area under a curve with parabolic segments. These parabolas fit the curve more accurately than straight lines, typically resulting in smaller errors.
Simpson's Rule uses the formula:\[ S_n = \frac{b-a}{3n} \left( f(a) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(b) \right), \]with \( n \) being an even number of intervals (since it applies parabolic approximation).
The error in Simpson's Rule is calculated as follows:\[ E_s = -\frac{(b-a)^5}{180n^4}f^{(4)}(c), \]where \( f^{(4)}(c) \) is the fourth derivative of the function evaluated at some point \( c \) in \([a,b]\). This formula demonstrates the potential for higher accuracy with fewer intervals, as Simpson's Rule captures more subtle curvature changes in the function.

Error Bound Calculations

Calculating error bounds is crucial to ensure our numerical integration is accurate enough for a given problem. Both the Trapezoidal Rule and Simpson's Rule have associated error equations, helping us find the necessary number of subintervals \( n \) to keep the error within a tolerable limit.
For the Trapezoidal Rule, the error bound is given by \[ |E_t| = \left| -\frac{(b-a)^3}{12n^2}f''(c) \right|, \]and for Simpson's Rule, it is\[ |E_s| = \left| -\frac{(b-a)^5}{180n^4}f^{(4)}(c) \right|. \]
By rearranging these formulas to solve for \( n \), we can determine the smallest \( n \) needed to ensure errors are within a specified threshold. This process often involves substituting the maximum possible value of the derivatives within the interval to get the worst-case error estimate. A typical goal is to ensure this error stays below a predefined small number, like 0.0001, to guarantee the approximation's reliability and precision.