Problem 24 - Calculus of a Single Variable

Question

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=\ln x, \quad 1 \leq x \leq 5 $$

Step-by-Step Solution

Answer

The arc length of the curve \(y = \ln x\) over the interval \(1 \leq x \leq 5\) approximately equals to the output given by the graphing utility when applied to the integral \(\int_1^5\sqrt{1+[1/x]^2}dx\). The exact numerical value would likely depend on the specific capabilities of your graphing utility.

Step 1: Sketch the Graph

First, sketch the graph of \(y = \ln x\). Show in the image the range of \(x\) values from 1 to 5.

Step 2: Understand Arc Length Formula

The arc length formula in calculus, for a function \(y = f(x)\) on the interval \([a, b]\), is given by \(L = \int_a^b\sqrt{1+[f'(x)]^2}dx\). Here, \(f'(x)\) represents the derivative of the function.

Step 3: Apply Arc Length Formula

The derivative of \(y = \ln x\) is \(f'(x) = 1/x\). Applying the arc length formula for the given function on the interval \([1, 5]\), we obtain the definite integral \(\int_1^5\sqrt{1+[1/x]^2}dx\).

Step 4: Recognize Integral Cannot be Directly Computed

Recognize that this integral cannot be evaluated directly using elementary functions and standard integration techniques.

Step 5: Approximate Arc Length using a Graphing Utility

Use a graphing utility with integration capabilities (like certain graphing calculators or computer software like Wolfram Alpha) to numerically solve the integral and approximate the arc length. Using such a utility, the approximate arc length gets calculated.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to find areas under a curve between two points on the x-axis. In our exercise, it helps us find the arc length of the curve \(y = \ln x\) from \(x = 1\) to \(x = 5\). The arc length formula \(L = \int_a^b\sqrt{1+[f'(x)]^2}dx\) enables us to determine the curve's length over a specific interval.
In this case, \(a\) and \(b\) are 1 and 5, respectively. The integral

• Accounts for the continuous nature of the curve.
• Provides the exact distance the curve covers on the interval.

Computing this particular integral directly is complex, which is why numerical methods might be necessary.

Derivative

The derivative is a measure of how a function changes as its input changes. For the function \(y = \ln x\), the derivative is \(f'(x) = 1/x\). Derivatives are essential in calculating arc lengths because they help in understanding the slope at every point along the curve.
In the formula \(L = \int_a^b\sqrt{1+[f'(x)]^2}dx\), the derivative helps evaluate the integrand:

• It shows how steep the curve is at any given moment.
• It allows the arc length formula to account for changes in the slope continuously.

By finding \(f'(x)\), we ensure that all variations in the curve are taken into account in the integral.

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions and solve complex calculations, like definite integrals, that might be challenging by hand. For the arc length of \(y = \ln x\) from 1 to 5, a graphing utility can be invaluable.
Here's why you might use one:

• It can numerically approximate integrals that are difficult to solve analytically.
• It provides visual insights into how the curve behaves over a specific interval.

Many educational tools, such as calculators and software, offer integration capabilities that streamline these processes, making learning more interactive and engaging.

Natural Logarithm

The natural logarithm, denoted \(\ln x\), is a mathematical function that emerges frequently in calculus and other mathematical fields. It is the logarithm to the base \(e\), where \(e\) is approximately 2.718. In our exercise, \(y = \ln x\) represents the curve whose arc length is being calculated.
Understanding \(\ln x\) is important because:

• It characterizes one particular type of exponential growth or decay.
• It possesses unique properties that make it distinct, such as the relationship \(d/dx(\ln x) = 1/x\).

Recognizing these features aids in calculating derivatives and further implications in calculus, making the understanding of \(\ln x\) crucial for the exercise.