Problem 6 - Essential Calculus

Question

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}, \quad[1,2] $$

Step-by-Step Solution

Answer

The integral for the arc length is nontrivial and generally requires numerical methods or software for accurate results. Without computing the exact integral, we cannot provide a numerical value for the arc length. However, the setup and understanding of how to approach the problem is outlined in this solution.

Step 1: Compute the Derivative of the Function

First, compute the derivative \(f^{\prime}(x)\) of the function \(f(x)=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}\). This results in \(f^{\prime}(x) = \frac{x^{3}}{2} - \frac{1}{2x^{3}}.\)

Step 2: Square the Derivative and Add One

Next, square the derivative and add one, yielding the integral's integrand: \[1+[f^{\prime}(x)]^{2}= 1+ \left(\frac{x^{3}}{2} - \frac{1}{2x^{3}}\right)^{2}.\] Simplification gives \[1+ \left(\frac{x^{3}}{2} - \frac{1}{2x^{3}}\right)^{2} = 1 + \frac{x^{6}}{4} - \frac{1}{4} + \frac{1}{4x^{6}} = \frac{x^{6}}{4} + \frac{1}{4x^{6}}.\]

Step 3: Compute the Square Root of the Expression

Then, compute the square root of the expression obtained in the previous step. This results in \[\sqrt{1+[f^{\prime}(x)]^{2}}=\sqrt{\frac{x^{6}}{4} + \frac{1}{4x^{6}}}.\]

Step 4: Integrate the Result Over the given interval

Finally, integrate this result over the given interval \([1, 2]\) using appropriate techniques. This integral is nontrivial and may require use of software or tables to yield the correct result. This will give the requested arc length.

Key Concepts

Derivative Calculation

To find the arc length of a curve defined by a function, we first need to calculate the derivative of the function. This process involves differentiating the function to find the rate at which the function's value changes with respect to its input variable. Calculating the derivative is a fundamental step in calculus, as it gives us the slope of the tangent line to the curve at any point. In our example, the function provided is \(f(x)=\frac{x^{4}}{8}+\frac{1}{4x^{2}}\). Here, we compute the derivative, denoted as \(f'(x)\). By using standard differentiation rules, we find the derivative \(f'(x) = \frac{x^{3}}{2} - \frac{1}{2x^{3}}\). This expression showcases the changes in the function with respect to changes in \(x\), thereby helping us understand the behavior of the curve.

Integrand Simplification

Once the derivative is calculated, the next step is to simplify the integrand, which is typically done by squaring the derivative and adding 1. The integrand simplifies the process of evaluating the arc length. For the given function, we perform the operation \(1+[f'(x)]^2\). Applying this to our previously calculated derivative, we have: - \(1+ \left(\frac{x^{3}}{2} - \frac{1}{2x^{3}}\right)^{2}\).By expanding and simplifying the terms, the expression becomes: - \(1 + \frac{x^{6}}{4} - \frac{1}{4} + \frac{1}{4x^{6}} = \frac{x^{6}}{4} + \frac{1}{4x^{6}}\).This simplification transforms the integrand into a manageable form for further steps in finding the arc length. Each term here represents a part of the arc length equation, making future calculations easier.

Integral Computation

The core concept of integral computation involves evaluating the integral of the simplified expression over the specified interval. In the context of arc length, the integration step is often non-trivial, requiring careful computation to determine the length of the curve.For the function we're investigating, the task involves integrating \(\sqrt{\frac{x^{6}}{4} + \frac{1}{4x^{6}}}\) over the interval \([1, 2]\). This part often involves:

• Application of integration techniques and formulas.
• Using numerical methods or software when necessary, especially for complex integrals.

Here, due to the complexity of the expression, computational tools or tables are beneficial. This integration process will ultimately yield the total arc length of the curve over the given interval. The result represents the accumulated total of how the curve stretches from \(x=1\) to \(x=2\).

Function Graph Analysis

Analyzing the function's graph is crucial in understanding the curve's shape and the arc length calculation. Graph analysis involves looking at the curve's behavior within the specified interval, which is critical for contextualizing the problem and the solution.For our function \(y=\frac{x^{4}}{8} + \frac{1}{4x^{2}}\), the graph reveals how the curve behaves between \(x=1\) and \(x=2\). This includes recognizing any turning points, symmetry, or regions of steep change, which can all impact the arc length. Such analysis helps in:

• Confirming the function behaves as predicted over the interval.
• Visualizing how the calculated arc length might manifest in practice.

By looking at the graph, students can gain intuition about the nature of the curve, aiding them in the solution and offering insight into how derivatives and integrals describe real-world shapes.