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Chapter 7: Problem 19

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. \(f(x)=\frac{1}{x}\) on [1,2]

Short Answer

Expert verified
The integral for the arc length is \( \int_{1}^{2} \sqrt{1 + \frac{1}{x^4}} \, dx \).

Step by step solution

01

Identify the Formula for Arc Length

To find the arc length of a function \( y = f(x) \) from \( x = a \) to \( x = b \), we use the formula for arc length: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
02

Differentiate the Function

Find the derivative of the function \( f(x) = \frac{1}{x} \). The derivative is \( f'(x) = -\frac{1}{x^2} \).
03

Square the Derivative

Square the derivative found in the previous step: \[ \left( \frac{dy}{dx} \right)^2 = \left( -\frac{1}{x^2} \right)^2 = \frac{1}{x^4} \]
04

Compute the Expression Inside the Square Root

Add 1 to the squared derivative: \[ 1 + \left( \frac{dy}{dx} \right)^2 = 1 + \frac{1}{x^4} \]
05

Set Up the Integral for Arc Length

Substitute the expression into the arc length formula: \[ L = \int_{1}^{2} \sqrt{1 + \frac{1}{x^4}} \, dx \] This integral represents the arc length of the function over the interval [1, 2].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
A definite integral represents the accumulation of quantities, which can be interpreted as the total "area" under a curve within a specified interval \( [a, b] \). Unlike an indefinite integral, which represents a family of functions and includes a constant of integration (\"C\"), the definite integral computes a numerical value for this total area.
  • Notation: A definite integral is noted \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the function being integrated over the interval \( [a, b] \).
  • Evaluation: It requires finding the antiderivative of \( f(x) \), then applying the "Fundamental Theorem of Calculus," which involves evaluating this antiderivative at the upper limit \( b \) and lower limit \( a \), and finding their difference. \[ F(b) - F(a) \]
  • Application: In the context of arc length, the definite integral helps us sum up the tiny line segments making up the curve from \( x = a \) to \( x = b \), giving us the total arc length.
By setting up the integral for arc length, we're prepared to calculate this length, although solving it is typically a more advanced step.
Derivatives
A derivative represents the rate at which a function changes at any given point. When we find the derivative of a function, we're essentially determining the slope or steepness of its curve at a specific point.
  • Definition: Mathematically, the derivative of a function \( f(x) \) is defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]
  • Notation: It can be denoted as \( f'(x) \), \( \frac{dy}{dx} \), or \( \frac{df}{dx} \), among others.
  • Purpose: Derivatives are used to find rates of change, tangents to curves, and optimizations among other things.
In our exercise, we find the derivative of \( f(x) = \frac{1}{x} \), which is \( f'(x) = -\frac{1}{x^2} \). Understanding this crucial step is necessary for proceeding with calculating the arc length, as it forms part of what's inside the square root in the arc length formula.
Differentiation
Differentiation is the process of finding a derivative, which provides critical insights into the behavior of functions. The rules and techniques for differentiation allow us to find derivatives systematically.
  • Basic Rule: The power rule is often used and states that for any function \( x^n \), the derivative is \( nx^{n-1} \).
  • Chain Rule: Necessary for composite functions \( f(g(x)) \), it states \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).
  • Product and Quotient Rules: These apply when differentiating functions that are multiplied together or divided. The quotient rule is used for \( \frac{u}{v} \) and is \( (\frac{u}{v})' = \frac{u'v - uv'}{v^2} \).
In our context, differentiation was used to calculate the derivative of \( f(x) = \frac{1}{x} \). The differentiation correctly applied resulted in \( f'(x) = -\frac{1}{x^2} \). Recognizing the importance of techniques like the quotient rule helps us arrive at accurate mathematical outcomes.

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Most popular questions from this chapter

A 6 ft cylindrical tank with a radius of \(3 \mathrm{ft}\) is filled with water, which has a weight density of \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\). The water is to be pumped to a point \(2 \mathrm{ft}\) above the top of the tank. (a) How much work is performed in pumping all the water from the tank? (b) How much work is performed in pumping \(3 \mathrm{ft}\) of water from the tank? (c) At what point is \(1 / 2\) of the total work done?

Find the total area enclosed by the functions \(f\) and \(g\). The functions \(f(x)=\cos (2 x)\) and \(g(x)=\sin x\) intersect infinitely many times, forming an infinite number of repeated, enclosed regions. Find the areas of these regions.

A force of \(7 \mathrm{~N}\) stretches a spring from a natural length of 11 \(\mathrm{cm}\) to \(21 \mathrm{~cm}\). How much work is performed in stretching the spring from a length of \(16 \mathrm{~cm}\) to \(21 \mathrm{~cm}\) ?

A region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes. The triangle with vertices (1,1),(1,2) and (2,1) . Rotate about: (a) the \(y\) -axis (b) \(x=1\) (c) the \(x\) -axis (d) \(y=2\)

Find the arc length of the function on the given interval. . \(f(x)=\sqrt{8} x\) on [-1,1]
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