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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 6.5.29a
Chapter 6, Problem 6.5.29a

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 = 1/x, for 1 ≤ x ≤ 10

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 interval: here, \( y = \frac{1}{x} \) and \( x \) ranges from 1 to 10.
Compute the derivative \( \frac{dy}{dx} \) of \( y = \frac{1}{x} \). Use the power rule or rewrite \( y = x^{-1} \) and differentiate.
Square the derivative \( \left(\frac{dy}{dx}\right)^2 \) and add 1 inside the square root to form the integrand: \[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
Write the integral for the arc length explicitly with the limits 1 to 10: \[ L = \int_1^{10} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] This integral can then be evaluated using a calculator or numerical methods.

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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 found using the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Arc Length of Parametric Curves

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = 1/x. The derivative measures the slope of the curve at each point and is essential for computing the integrand √(1 + (dy/dx)^2).
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Derivatives of Other Trig Functions

Definite Integral Evaluation

After setting up the integral for arc length, evaluating it over the interval [1, 10] gives the total length. This may require simplification or numerical methods, such as using a calculator, especially when the integral cannot be expressed in elementary functions.
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Definition of the Definite Integral
Related Practice
Textbook Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.


a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.


C′(x)=200−0.05x

Textbook Question

Region R is revolved about the line x=4 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point y in [1, 3]?

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

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Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


a. Use the shell method to write an integral for the volume of the torus.