Skip to main content
Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.5.23a
Chapter 6, Problem 6.5.23a

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 
y = ln x, for 1≤x≤4

Verified step by step guidance
1
Recall the formula for the arc length of a curve given by a function \( y = f(x) \) on the interval \( [a, b] \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Identify the function and the interval: here, \( y = \ln x \) and the interval is \( 1 \leq x \leq 4 \).
Compute the derivative \( \frac{dy}{dx} \) of \( y = \ln x \): \[ \frac{dy}{dx} = \frac{1}{x} \]
Substitute \( \frac{dy}{dx} \) into the arc length formula: \[ L = \int_1^4 \sqrt{1 + \left(\frac{1}{x}\right)^2} \, dx = \int_1^4 \sqrt{1 + \frac{1}{x^2}} \, dx \]
Simplify the expression under the square root: \[ \sqrt{1 + \frac{1}{x^2}} = \sqrt{\frac{x^2 + 1}{x^2}} = \frac{\sqrt{x^2 + 1}}{x} \] So the integral becomes: \[ L = \int_1^4 \frac{\sqrt{x^2 + 1}}{x} \, dx \]

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
Recommended video:
06:29
Arc Length of Parametric Curves

Derivative of the Natural Logarithm Function

For y = ln(x), the derivative dy/dx is 1/x. Understanding this derivative is essential to substitute into the arc length formula, as it determines the slope of the curve at each point.
Recommended video:
05:18
Derivative of the Natural Logarithmic Function

Simplifying the Integral Expression

After substituting dy/dx into the arc length formula, the integral often requires algebraic simplification before evaluation. Simplifying √(1 + (1/x)²) to √(1 + 1/x²) = √((x² + 1)/x²) = √(x² + 1)/x helps in setting up the integral for numerical or calculator-based evaluation.
Recommended video:
6:36
Simplifying Trig Expressions
Related Practice
Textbook Question

Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


a. What is the SAV ratio of a cube with side lengths a?

Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

a. What is the radius of a cylindrical shell at a point x in [0, 2]?

Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

1
views
Textbook Question

A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.

a. How much work is required to compress the spring and lift the mass 0.5 m?

1
views
Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 6−2t on [0, 5]; s(0)=0

1
views