Problem 23 - Calculus: Early Transcendentals

Question

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=\ln x \text { on }[1, e] $$

Step-by-Step Solution

Answer

The integral is \[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \,dx \].

Step 1: Understand the Arc Length Formula

The arc length \( L \) of a function \( y = f(x) \) over an interval \([a, b]\) can be computed using the integral formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \,dx \] where \( \frac{dy}{dx} \) is the derivative of the function \( f(x) \).

Step 2: Compute the Derivative of the Function

The given function is \( f(x) = \ln x \). To find the derivative, we use the rule that \( \frac{d}{dx} (\ln x) = \frac{1}{x} \). Thus, \( \frac{dy}{dx} = \frac{1}{x} \).

Step 3: Substitute the Derivative into the Arc Length Formula

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

Step 4: Simplify the Integral Expression

Simplify the expression under the square root: \( \sqrt{1 + \frac{1}{x^2}} \) remains as is since further simplification isn't straightforward. Thus, the integral is set up as: \[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \,dx \] This expression is the correct integral for the arc length of the function \( f(x) = \ln x \) on the interval \([1, e]\).

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that focuses on the accumulation of quantities. While differential calculus deals with rates of change, integral calculus deals with the quantity of change over intervals. One of the exciting applications of integral calculus is in determining the arc length of a curve.

• To find the arc length of a function, we set up an integral.
• This integral sums small distances along the curve to get the total length.

In our case, the function given is a natural logarithm, and we're focusing on the interval \([1, e]\). The integral accumulates these small distances to give us the length of the curve formed by \((x) = \ln x\). This process involves using the arc length formula, which is an integral that provides the sum of infinitesimal line segments along the curve.

Derivative

The derivative of a function is a key concept in calculus that measures how the function changes as its input changes. In simpler terms, it tells you the slope of the function at any given point.

• For our function \(f(x) = \ln x\), the derivative, denoted as \(\frac{dy}{dx}\), tells us how steep the graph is at any point \(x\).
• The derivative of \(\ln x\) is \(\frac{1}{x}\).

Knowing the derivative helps us set up the integral for the arc length. We plug this derivative into the arc length formula to account for the changing slope of the curve. The derivative, therefore, is not just a tool for finding slopes but also a crucial piece in determining our integral's accuracy.

Natural Logarithm

The natural logarithm function, usually written as \(\ln x\), is a fundamental mathematical function. It features prominently in many area of calculus, as it is the inverse of the exponential function \(e^x\).

• The natural logarithm measures growth patterns. It's particularly useful in calculus because of its properties.
• The derivative of \(\ln x\) is \(\frac{1}{x}\), which simplifies many calculus problems.

In our exercise, we are calculating the arc length of \(\ln x\). Knowing its behavior and derivative is crucial to set up the integral correctly. The natural logarithm is smooth and predictable, which makes it a little less overwhelming to deal with in calculus problems.

Arc Length Formula

The arc length formula is a fundamental tool in integral calculus used to find the length of curves. It is given by the formula:\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \,dx \]This formula combines our knowledge of derivatives with integration.

• First, we compute the derivative \(\frac{dy}{dx}\) of the function.
• Then, we plug this derivative into the formula under the square root.

The square root of \(1 + \left(\frac{dy}{dx}\right)^2\) is used to adjust for the slope of the curve over the interval. This ensures that as the curve changes direction, our length calculation follows the actual path and not just the straight-line distance. For the function \(f(x) = \ln x\) over \[1, e\], the arc length formula results in the integral:\[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \,dx \] The formula might look complex, but it elegantly encapsulates everything needed to find the curve's length.