Question

Determine an \(n\) so that the Trapezoidal Rule will approximate the integral with an error \(E_{n}\) satisfying \(\left|E_{n}\right| \leq 0.01 .\) Then, using that \(n\), approximate the integral. $$ \int_{1}^{3} \frac{1}{1+x} d x $$

Short answer

To have an error \(|E_n| \leq 0.01\), choose \(n = 5\). The integral is approximately \(0.695\).

Step-by-step solution

Write Down the Trapezoidal Rule Error Formula

The error for the Trapezoidal Rule for approximating the integral \( \int_{a}^{b} f(x) \, dx \) is given by \( E_n = -\frac{(b-a)^3}{12n^2} f''(c) \), where \( a \leq c \leq b \). This formula helps us determine the number of trapezoids, \(n\), needed for a desired accuracy.

Find the Second Derivative of the Function

The function for the integral is \( f(x) = \frac{1}{1+x} \). We first differentiate \( f(x) \) to get \( f'(x) = -\frac{1}{(1+x)^2} \). Then, differentiate again to find \( f''(x) = \frac{2}{(1+x)^3} \).

Determine the Maximum of \(f''(x)\) on \([1, 3]\)

The second derivative is \( f''(x) = \frac{2}{(1+x)^3} \). This function decreases as \(x\) increases on the interval \([1,3]\). Therefore, the maximum value of \( f''(x) \) is at \( x=1 \) and is \( f''(1) = \frac{2}{8} = \frac{1}{4} \).

Set Up the Error Inequality for \( n \)

To find \(n\) such that \(|E_n| \leq 0.01\), use the error formula: \[ \left| -\frac{(b-a)^3}{12n^2} \cdot \max |f''(x)| \right| \leq 0.01 \]. Substituting the known values, \((b-a)=2\), and \(\max |f''(x)| = \frac{1}{4}\), the inequality becomes \[ \frac{8}{48n^2} \leq 0.01 \].

Solve the Inequality for \( n \)

Simplify the inequality \( \frac{1}{6n^2} \leq 0.01 \) to \( 1 \leq 0.06n^2 \) then \( n^2 \geq \frac{1}{0.06} \). Thus, \( n^2 \geq 16.67 \), and \( n \geq \sqrt{16.67} \approx 4.08 \). Hence, we choose \( n = 5 \) to ensure the condition is satisfied.

Approximate the Integral Using \( n = 5 \)

For \( n = 5 \), the interval \([1, 3]\) is divided into 5 equal subintervals, each of width \( \Delta x = \frac{2}{5} = 0.4 \). Write down the Trapezoidal Rule formula: \[ T_5 = \frac{\Delta x}{2} [f(1) + 2f(1.4) + 2f(1.8) + 2f(2.2) + 2f(2.6) + f(3)] \]. Calculate each term \( f(x) \) and plug into this formula.

Perform the Calculations

Calculate each \( f(x) \) term:- \( f(1) = \frac{1}{2} \), \( f(1.4) = \frac{1}{2.4} \approx 0.4167 \), \( f(1.8) = \frac{1}{2.8} \approx 0.3571 \)- \( f(2.2) = \frac{1}{3.2} \approx 0.3125 \), \( f(2.6) = \frac{1}{3.6} \approx 0.2778 \), \( f(3) = \frac{1}{4} = 0.25 \).Insert these into the formula and compute:\[ T_5 \approx \frac{0.4}{2} \left(0.5 + 2(0.4167) + 2(0.3571) + 2(0.3125) + 2(0.2778) + 0.25\right) = 0.695 \]

Conclusion of the Integral Approximation

The approximate value of the integral \( \int_{1}^{3} \frac{1}{1+x} \, dx \) using the Trapezoidal Rule with \( n=5 \) is \( \approx 0.695 \). This approximation satisfies the error condition \( |E_n| \leq 0.01 \).

Key concepts

Integral Approximation

Approximating the value of an integral is key in situations where finding the exact value analytically is complex or impractical. One common method used for this purpose is the Trapezoidal Rule. This technique involves approximating the area under a curve as a series of trapezoids and summing their areas. For the integral \( \int_{1}^{3} \frac{1}{1+x} \ dx \), this method allows us to estimate the area by dividing the curve into sections between 1 and 3. In our exercise, we've determined an appropriate number of intervals (or trapezoids), \( n = 5 \), to achieve an accurate approximation within the specified error bounds.

Error Analysis

Understanding the error inherent in numerical approximation is critical. Each approximation method comes with its own potential for error, and the Trapezoidal Rule error can be quantified using the formula: \[ E_n = -\frac{(b-a)^3}{12n^2} f''(c) \], where \( a \leq c \leq b \). For our problem, we analyzed the second derivative of the function to determine the maximum error that could occur in our chosen interval. We found that the maximum \( f''(x) = \frac{2}{(1+x)^3} \) takes the value \( \frac{1}{4} \) at \( x = 1 \). Using this information, our calculation ensured that the error of approximation stays under the desired threshold of 0.01 by choosing \( n = 5 \).

Calculus

Calculus provides the mathematical foundation for analyzing continuous change. When dealing with integrals, calculus allows us to understand areas under curves. The function given, \( f(x) = \frac{1}{1+x} \), describes a curve that doesn't yield easily to simple algebraic solution methods. By applying derivatives, we found that \( f'(x) = -\frac{1}{(1+x)^2} \) and \( f''(x) = \frac{2}{(1+x)^3} \). These derivatives are important for assessing the behavior of the function and thus crucial for both understanding the function's shape and for calculating maximum error in the approximation.

Numerical Integration

Numerical integration is a powerful tool for estimations. Unlike analytical integration, which seeks the exact area under a curve, numerical methods like the Trapezoidal Rule are practical approaches to approximate the integral with high accuracy. This is especially useful for functions that are difficult or impossible to integrate analytically. For our specific example \( \int_{1}^{3} \frac{1}{1+x} \ dx \), numerical methods provided an effective approximation using 5 intervals, achieving a satisfying and well-bounded result of approximately 0.695. This approach not only demonstrates the utility of numerical integration in calculus but also showcases its application in real-world scenarios where precision and efficiency are paramount.