Problem 62 - Calculus for Biology and Medicine

Question

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=\ln x, 1 \leq x \leq e\)

Step-by-Step Solution

Answer

Set up the integral as \( \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \).

Step 1: Identify the Formula for Arc Length

The arc length of a curve from point \(x = a\) to \(x = b\) can be found using the formula: \\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]\We need to apply this formula to the given function \(y = \ln x\).

Step 2: Differentiate the Function

To apply the arc length formula, we need to find \(\frac{dy}{dx}\) for the function \(y = \ln x\). The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\).

Step 3: Set Up the Integrand

Substitute \(\frac{dy}{dx} = \frac{1}{x}\) into the arc length formula: \\[ L = \int_{1}^{e} \sqrt{1 + \left( \frac{1}{x} \right)^2} \, dx \]\Simplify the expression under the square root: \\[ \sqrt{1 + \left( \frac{1}{x} \right)^2} = \sqrt{1 + \frac{1}{x^2}} \]

Step 4: Write Down the Integral

Now, set up the integral for arc length as: \\[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \]\This integral is set up but not evaluated, as required by the exercise.

Key Concepts

Integration

Integration is a powerful mathematical tool that helps find quantities like areas under curves, volumes, and arc lengths. When we talk about setting up an integral for a curve's arc length, what we're doing is calculating the total distance along the curve from one point to another.

For arc length, integration involves using a specific formula:

• It takes the derivative of the function, squares it, adds 1, and then takes the square root.
• This expression under the square root is then integrated over the interval of interest, in this case, from 1 to \(e\) for the function \(y = \ln x\).

By setting up the integral \( L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \), we've determined how far the curve travels from \(x = 1\) to \(x = e\) along the natural logarithmic curve.

This integral represents our journey along the curve, visualizing how the slope changes and affects the arc length. Integration helps us sum up these little distances to find the complete arc length of the curve.

Differentiation

Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. Think of it as the slope of a function at any given point.

In the context of arc length, differentiation is essential for determining the integrand. For the logarithmic function \(y = \ln x\), the derivative is \(\frac{dy}{dx} = \frac{1}{x}\).

Here's why differentiation is crucial:

• Calculating the derivative helps us understand how steep or flat the curve is at any point.
• Substituting the derivative into the arc length formula lets us directly see how the curve's slope affects the total arc length.

Performing differentiation gives us the specific rate of change, which plays an integral role in calculating the arc length. Without knowing \(\frac{1}{x}\), we couldn't have completed the setup for \(L = \int_{1}^{e} \sqrt{1 + \left( \frac{1}{x} \right)^2} \, dx \). It links differentiation and integration in a bid to understand the full arc length.

Logarithmic Functions

Logarithmic functions, such as \(y = \ln x\), are inverse operations to exponential functions and describe very common real-world phenomena, from sound intensity to interest rates.

Understanding these functions is key to many mathematical applications:

• The natural logarithm, denoted as \(\ln x\), grows slowly and continuously, a curve we examined between 1 and \(e\).
• They possess unique properties, such as the derivative \(\frac{1}{x}\), that simplify when computing integrals.

When dealing with arc length problems featuring logarithmic functions:

• Recognize the curvy yet steadily increasing nature of \(\ln x\).
• This smooth increase blends into the integration process, yielding a clearer picture of how long the curve stretches along its domain.

The characteristic of gradual change in the logarithmic curve means even complex curves like those of \(y = \ln x\) can be tackled confidently through differentiation and integration. These principles open a wide range of possibilities for mathematicians and scientists alike.