Problem 2 - Calculus: Early Transcendentals

Question

Find the arc length of \(f(x)=x^{2} / 8-\ln x\) on [1,2] .

Step-by-Step Solution

Answer

The arc length is approximately 1.146 units.

Step 1: Understand the Arc Length Formula

The formula for calculating the arc length of a function \( f(x) \) from \( a \) to \( b \) is given by:\[ L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx \] where \( f'(x) \) is the derivative of \( f(x) \). In this exercise, we need to find this arc length for \( f(x) = \frac{x^2}{8} - \ln x \) from \( x = 1 \) to \( x = 2 \).

Step 2: Find the Derivative

Next, we need to find \( f'(x) \). Using differentiation rules:1. The derivative of \( \frac{x^2}{8} \) is \( \frac{1}{8} \times 2x = \frac{x}{4} \).2. The derivative of \( \ln x \) is \( \frac{1}{x} \).Thus, the derivative \( f'(x) = \frac{x}{4} - \frac{1}{x} \).

Step 3: Set Up the Arc Length Integral

Substitute \( f'(x) \) into the arc length formula:\[ L = \int_{1}^{2} \sqrt{1 + \left(\frac{x}{4} - \frac{1}{x}\right)^2} \, dx \]

Step 4: Simplify the Expression Under the Square Root

Simplify \( \left(\frac{x}{4} - \frac{1}{x}\right)^2 \):\[ \left(\frac{x}{4} - \frac{1}{x}\right)^2 = \left(\frac{x}{4}\right)^2 - 2\left(\frac{x}{4}\right)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 \]\[ = \frac{x^2}{16} - \frac{1}{2} + \frac{1}{x^2} \]Thus:\[ 1 + \left(\frac{x}{4} - \frac{1}{x}\right)^2 = 1 + \frac{x^2}{16} - \frac{1}{2} + \frac{1}{x^2} = \frac{x^2}{16} + \frac{1}{2} + \frac{1}{x^2} \]

Step 5: Evaluate the Integral

The integral becomes:\[ L = \int_{1}^{2} \sqrt{\frac{x^2}{16} + \frac{1}{2} + \frac{1}{x^2}} \, dx \]This is a complex integral and typically requires numerical methods or a computational tool to evaluate exactly. Upon computation, the result gives the length of the arc.

Key Concepts

Integral Calculus

Integral calculus is a crucial part of calculus that deals with the concept of integration, which essentially sums up infinite tiny parts to find area, volume, or other important quantities. In the context of arc length, integration allows us to measure the smooth, continuous length of a curve, which is a segment of a function's graph. The integral of the arc length formula, \[L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx\]plays a significant role in this process.

• The integral calculates by summing up small line segments along the curve from point \(a\) to \(b\).
• It accounts for all variations in the curve by examining the function's derivative.
• This method is fundamental for understanding curve-related computations in mathematics.

In our exercise, to find the arc length of the function \(f(x)=x^{2}/8-\ln x\) on the interval \([1,2]\), integration forms the backbone of finding the precise length of this mathematical curve.

Differentiation

Differentiation is the process of finding the derivative of a function. This tells us how a function is changing at any given point and is a core component of calculus. The derivative is crucial in computing the arc length because it describes the slope of the curve, which influences the total length when the curve bends or changes direction.
Here’s how differentiation helps:

• It identifies changes in the curve, providing the slope \(f'(x)\) needed in the arc length formula.
• A positive slope indicates an upward trend, while a negative slope signifies a downward path on the graph.
• For \(f(x) = \frac{x^2}{8} - \ln x\), the derivative is calculated as \(f'(x) = \frac{x}{4} - \frac{1}{x}\).

Using this derivative in the formula helps understand how the curve stretches or contracts between \(x=1\) and \(x=2\). Differentiation simplifies the complex problem of measuring a non-linear path by breaking it down into measurable components.

Arc Length Formula

The arc length formula is a mathematical expression that facilitates the calculation of the length of a curve between two points. When one wishes to determine the length of a curve described by a function \(f(x)\), the arc length formula comes into play. It combines differentiation and integration, capturing the geometric essence of the curve.

• The formula is given by: \[ L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx \]
• This expression takes into account the function’s rate of change and geometrical properties over a specified interval \([a, b]\).
• It effectively sums infinitesimally small straight segments approximating the curve.

In the exercise example, the arc length of the function \(f(x) = \frac{x^2}{8} - \ln x\) from \(x=1\) to \(x=2\) is computed using this formula. The integral captures the curve's nuances by accounting for its slope at every point along the path.