Question

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

Short answer

The integral is \( \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \).

Step-by-step solution

Understand the Arc Length Formula

The formula for the arc length of a function \( f(x) \) from \( x=a \) to \( x=b \) is given by: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2 } \, dx \]The function provided is \( f(x) = \ln x \). The interval is \([1, e]\).

Differentiate the Function

Find the derivative of \( f(x) = \ln x \).The derivative is \( \frac{df}{dx} = \frac{1}{x} \).

Set Up the Integrand

Substitute \( \frac{df}{dx} \) into the arc length formula. The integrand becomes:\[ \sqrt{1 + \left( \frac{1}{x} \right)^2 } = \sqrt{1 + \frac{1}{x^2}} \]

Set Up the Integral

Now, set up the required integral from \( a = 1 \) to \( b = e \):\[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \]

Final Integral Expression

The integral to compute the arc length of \( f(x)=\ln x \) over the interval \([1, e]\) is:\[ L = \int_{1}^{e} \sqrt{1 + \frac{1}{x^2}} \, dx \]

Key concepts

Integration

Integration is a fundamental concept in calculus, involving the calculation of an integral. It is often used to find areas, volumes, and in this case, arc lengths. The arc length of a curve can be expressed with an integral formula, which requires finding the length of the curve between two points. The formula to compute arc length is given by:\[L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2 } \, dx\]Here, \(L\) represents the arc length, and the limits of integration \(a\) and \(b\) represent the interval over which the function \(f(x)\) is defined, in this case from \(x=1\) to \(x=e\). This integral takes into account every tiny segment or piece of the curve on this interval, adding them up to form the total length.Understanding integration in the context of arc length is about setting up and understanding this integral expression. Though the question emphasizes setup and not evaluation, mastering this process is crucial to solve calculus problems related to lengths of curves.

Derivative

The concept of a derivative is central in calculus, representing the rate of change of a function. When considering the arc length, the derivative helps us determine how steep or flat a curve is at any given point. For a function \(f(x)\), the derivative \(\frac{df}{dx}\) provides the slope of the tangent line to the curve.In the case of \(f(x) = \ln x\), the derivative is computed as:\[\frac{df}{dx} = \frac{1}{x}\]This formula tells us that the slope decreases as \(x\) increases. Such derivatives are important because they signify how the function grows or declines. While setting up the arc length integral, this derivative becomes part of the expression we integrate over the interval.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is an important mathematical function used in many fields. It is the inverse operation of the exponential function \(e^x\). The natural logarithm has a distinctive feature: it increases gradually as \(x\) increases and is defined for positive real numbers. When differentiating \(\ln x\), the result is \(\frac{1}{x}\), which represents the rate of change of the logarithmic function. Understanding \(\ln x\) and its derivative is crucial when dealing with mathematical problems involving growth, decay, and continuous compounding. The natural logarithm is often used in physics, probability, and many real-world logarithmic calculations.

Integral Setup

Setting up the integral properly is essential when working with arc lengths. Here is what you should consider:

• Function and Interval Identification: Clearly identify the function, \(f(x)=\ln x\), and the interval \([1, e]\).
• Calculate Derivative: Compute the derivative of the function, \(\frac{df}{dx} = \frac{1}{x}\), which will be used in the formula.
• Arc Length Formula Substitution: Use the formula \(L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2 } \, dx\) by substituting the derivative.
• Integrand Simplification: Simplify the integrand to \(\sqrt{1 + \frac{1}{x^2}}\).
• Set Limits: Ensure the bounds of the integral correctly reflect the problem: \(\int_{1}^{e}\).

By methodically setting up the integral, you lay the foundation for accurate calculations. This setup is indispensable when you later evaluate or further explore the integral using different methods.