Problem 28 - Calculus Early Transcendentals

Question

Find the arc length of the following curves by integrating with respect to \(y\) $$y=\ln (x-\sqrt{x^{2}-1}), \text { for } 1 \leq x \leq \sqrt{2}$$

Step-by-Step Solution

Answer

Question: Find the arc length of the curve given by the function \(y=\ln (x-\sqrt{x^{2}-1})\) for \(1 \leq x \leq \sqrt{2}\). Answer: The arc length of the curve can be approximated by numerically computing the integral: $$L = \int_{-\infty}^{\ln(\sqrt{2}-1)} \sqrt{1 + \left( x - \sqrt{x^2 - 1} \right)^2} dy$$

Step 1: Find the expression for arc length with respect to \(y\)

The arc length of a curve, given by the function \(y=f(x)\) and for \(x\) values in the range \([a, b]\), can be expressed as: $$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$ To express it in terms of \(y\), we'll need to find \(\frac{dx}{dy}\) and change the limits of integration accordingly.

Step 2: Differentiate the given function with respect to \(x\) and find \(\frac{dx}{dy}\)

Given that \(y = \ln (x-\sqrt{x^2 - 1})\), differentiate with respect to \(x\): $$\frac{dy}{dx} = \frac{1}{x - \sqrt{x^2 - 1}} \cdot \frac{d}{dx} (x - \sqrt{x^2 - 1})$$ $$\frac{dy}{dx} = \frac{1}{x - \sqrt{x^2 - 1}} \cdot \left(1 - \frac{x}{\sqrt{x^2 - 1}}\right)$$ Now find \(\frac{dx}{dy}\) by taking the reciprocal: $$\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = x - \sqrt{x^2 - 1}$$

Step 3: Change the limits of integration in terms of \(y\)

The given limits for \(x\) are: \(1 \leq x \leq \sqrt{2}\). We need to find the corresponding limits for \(y\). For \(x=1\), we have: $$y = \ln(1-\sqrt{1^2-1}) = \ln(0) = -\infty$$ For \(x=\sqrt{2}\), we have: $$y = \ln(\sqrt{2}-\sqrt{(\sqrt{2})^2-1}) = \ln(\sqrt{2}-1)$$ The range of \(y\) is therefore \([-\infty, \ln(\sqrt{2}-1)]\).

Step 4: Calculate the arc length by integrating with respect to \(y\)

Substitute \(\frac{dx}{dy}\) into the arc length expression and apply the new integration limits: $$L = \int_{-\infty}^{\ln(\sqrt{2}-1)} \sqrt{1 + \left( x - \sqrt{x^2 - 1} \right)^2} dy$$ The above integral is complex and difficult to solve analytically. However, you can use numerical methods or a calculator to find an approximate value for the arc length of the curve.

Key Concepts

Integration with Respect to y

When calculating the arc length of curves, we sometimes integrate with respect to \(y\) rather than \(x\). This approach is especially useful when the function is easier to work with in terms of \(y\) or when the integration bounds are cleaner. In the given exercise, we start by noting the traditional arc length formula is:

• \(L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\)

However, to integrate with respect to \(y\), we consider the inverse function approach. First, we find \(\frac{dx}{dy}\), the reciprocal of \(\frac{dy}{dx}\). The transformation from \(x\)-limits to \(y\)-limits is achieved by considering the function values at these intervals. Changing integration from \(\int dx\) to \(\int dy\) involves calculating the new expression for each variable and adapting the integral accordingly. This is a crucial step in setting up the integral correctly when examining functions expressed in different terms.

Differentiation

Differentiation plays a key role in finding the rate of change or the slope of a curve. For computing arc length, differentiation helps us obtain \(\frac{dy}{dx}\), the derivative of the function concerning \(x\). Here's how it applies in our problem:
Given \(y = \ln (x - \sqrt{x^2 - 1})\), the differentiation process involves applying chain rule and techniques for differentiating logarithmic and square root functions:

• The logarithmic derivative: \(\frac{d}{dx} \left(\ln (x - \sqrt{x^2 - 1})\right)\)
• The chain rule for the composite part: \(\frac{d}{dx} (x - \sqrt{x^2 - 1})\)

The reciprocal of this derivative gives \(\frac{dx}{dy}\). Understanding each differentiation step in the context of arc length aids in transforming the problem appropriately, and ensures correct computation when switching between \(x\) and \(y\). These steps are necessary because they allow you to manipulate the function into a form that suits the new variable of integration, ensuring accurate evaluation.

Limits of Integration

The limits of integration determine the range over which we calculate the arc length. When integrating with respect to \(y\), recalibrating limits from \(x\) to \(y\) is necessary. For instance, given the function \(y = \ln (x - \sqrt{x^2 - 1})\), we find the limits by evaluating the function at given \(x\) values. For this problem:

• For \(x = 1\), \(y = \ln(1 - \sqrt{1^2 - 1}) = \ln(0) = -\infty\).
• For \(x = \sqrt{2}\), \(y = \ln(\sqrt{2} - \sqrt{2 - 1}) = \ln(\sqrt{2} - 1)\).

These transformations ensure the integral is evaluated over the correct interval. When moving from \(x\) to \(y\), always verify that limits are adjusted accurately to reflect the correct span of the function. This step is vital in maintaining consistency across transformations, ensuring the accuracy of the integral's calculation and reflecting the actual arc length along the specified region of the curve.