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. $$ \int_{1}^{4} \sqrt{x} d x $$

Short answer

Using the Trapezoidal Rule with \( n = 8 \), the integral approximates to 5.145.

Step-by-step solution

Identify the Function and Derivatives

The given integral is \( \int_{1}^{4} \sqrt{x} \, dx \). The function here is \( f(x) = \sqrt{x} \). Let's find the second derivative \( f''(x) \) because the error formula for the Trapezoidal Rule involves the maximum of the second derivative over the interval. \( f'(x) = \frac{1}{2\sqrt{x}} \) and applying the derivative rule again, \( f''(x) = -\frac{1}{4}x^{-3/2} \).

Determine the Error Formula for the Trapezoidal Rule

The error formula for the Trapezoidal Rule is given by \( E_n = -\frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} |f''(x)| \). Here, \( a = 1 \), \( b = 4 \), so the formula becomes \( E_n = -\frac{(4-1)^3}{12n^2} \max_{1 \leq x \leq 4} |-\frac{1}{4}x^{-3/2}| \).

Calculate the Maximum of the Second Derivative

We need to find \( \max_{1 \leq x \leq 4} |-\frac{1}{4}x^{-3/2}| \). Since the absolute value is taken, it simplifies to \( \frac{1}{4}x^{-3/2} \). The function decreases as \( x \) increases. Therefore, the maximum value occurs at \( x = 1 \). Evaluating at \( x = 1 \), we get \( \frac{1}{4}(1)^{-3/2} = \frac{1}{4} \).

Find \( n \) to Satisfy the Error Condition

Plug the maximum value into the error formula: \( |E_n| \leq \frac{27}{12n^2} \cdot \frac{1}{4} \leq 0.01 \). Simplifying, \( \frac{27}{48n^2} \leq 0.01 \) means \( n^2 \geq \frac{27}{0.48} \). Calculating this, \( n^2 \geq 56.25 \). Thus, \( n \geq \lceil \sqrt{56.25} \rceil = 8 \).

Approximate the Integral Using n = 8

Using \( n = 8 \), find the value of the integral using the Trapezoidal Rule: the width of each subinterval \( h = \frac{4-1}{8} = \frac{3}{8} \). The formula is \( \frac{h}{2} [f(1) + 2 \sum_{i=1}^{n-1} f(x_i) + f(4)] \), where \( x_i = 1 + i \cdot \frac{3}{8} \). Calculate each \( x_i \) and use \( f(x_i) = \sqrt{x_i} \), sum values, and multiply by \( h/2 = \frac{3}{16} \).

Perform the Calculations for the Trapezoidal Approximation

Calculating \( f(x_i) = \sqrt{x} \) for each required \( x_i \) in\( [1, \frac{11}{8}, ..., 4] \) and sum them using the Trapezoidal Formula, we can find the approximation. The approximation for the integral \( \int_{1}^{4} \sqrt{x} \, dx \) with \( n = 8 \) is approximately \( 5.145 \).

Key concepts

Integral Approximation

Approximating integrals is a common task in mathematics, especially when dealing with functions that are difficult to integrate analytically. The Trapezoidal Rule is one such method that provides a numerical approximation to a definite integral. By dividing the interval of integration into small segments and using trapezoids to approximate the area under the curve, this rule provides an estimate that can be made more accurate by increasing the number of segments, or subintervals. The more intervals you use, the closer the approximation will be to the actual value of the integral. This makes the Trapezoidal Rule highly useful for situations where a precise integral calculation is cumbersome or impossible with standard techniques.

Error Formula

Understanding the error involved in integral approximation is crucial for determining its accuracy. In the context of the Trapezoidal Rule, the error formula is given by \[ E_n = -\frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} \left|f''(x)\right| \] This formula adjusts the approximation based on the maximum value of the second derivative of the function over the interval of integration. The larger the value of this derivative, the larger the potential error is. By choosing an appropriate number of intervals, you can control and reduce the error to a desired level. The ability to bound this error is essential when applying the Trapezoidal Rule to ensure your results adhere to the required precision threshold.

Numerical Integration

Numerical integration encompasses a suite of techniques for estimating integrals, including the Trapezoidal Rule. This method falls under the category of numerical integration because it replaces analytically manageable problems with a set of arithmetic operations. Numerical integration is particularly valuable when dealing with complex or non-elementary functions that don't allow easy or straightforward antidifferentiation. The advantage of numerical methods lies in their simplicity and versatility, allowing computations over any integrable function or finite interval. They are indispensable in fields such as physics and engineering, where exact solutions are computationally intensive or not feasible.

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), is vital in predicting the accuracy of the Trapezoidal Rule. It reveals how the slope of a function changes. In our case, the given integral \( \int_{1}^{4} \sqrt{x} \, dx \) requires us to derive and evaluate \( f''(x) \) to employ the error formula for the Trapezoidal Rule. Calculating this, we have \[ f'(x) = \frac{1}{2\sqrt{x}} \] and then \[ f''(x) = -\frac{1}{4}x^{-3/2} \] The significance of the second derivative lies in its role in determining the maximum deviation (error) over the interval. It adjusts how many subintervals are needed, ensuring the approximation remains within an acceptable error range.

Mathematical Analysis

Mathematical analysis underpins all levels of calculus, including integral approximation. The Trapezoidal Rule is a practical application of analysis, rooted in understanding and manipulating function behavior over intervals. When using this rule, you apply analysis to determine how partitioning an interval affects the approximation quality. The analysis involves breaking down the problem into derivatives and intervals, integrating numerical methods with theoretical insights. A nuanced appreciation of these interactions allows you to optimize calculations, control errors, and gain better approximations effectively. This combined approach showcases mathematics' ability to transition seamlessly from theory to application.