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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 6.5.27b
Chapter 6, Problem 6.5.27b

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


b. If necessary, use technology to evaluate or approximate the integral.
y = cos 2x, for 0 ≤ x ≤ π

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Recall the formula for the arc length of a curve defined by a function \( y = f(x) \) from \( x = a \) to \( x = b \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Identify the function and the interval: here, \( y = \cos(2x) \) and the interval is \( 0 \leq x \leq \pi \).
Compute the derivative \( \frac{dy}{dx} \) of \( y = \cos(2x) \) using the chain rule: \[ \frac{dy}{dx} = -2 \sin(2x) \]
Substitute \( \frac{dy}{dx} \) into the arc length formula to get the integral: \[ L = \int_0^{\pi} \sqrt{1 + (-2 \sin(2x))^2} \, dx = \int_0^{\pi} \sqrt{1 + 4 \sin^2(2x)} \, dx \]
Since this integral is not straightforward to solve analytically, use a calculator or appropriate technology to approximate the value of the integral over the interval \( [0, \pi] \).

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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 given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the distance along the curve by summing infinitesimal line segments, requiring the derivative of the function.
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Arc Length of Parametric Curves

Derivative of the Function

To find the arc length, you must compute dy/dx, the derivative of y with respect to x. For y = cos(2x), use the chain rule: dy/dx = -2 sin(2x). This derivative is then squared and used inside the arc length integral.
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Derivatives of Other Trig Functions

Use of Technology for Integration

Some integrals, like the arc length integral for y = cos(2x), may not have simple antiderivatives. Technology such as graphing calculators or computer algebra systems can approximate or evaluate these integrals numerically, providing practical solutions.
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Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?

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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.

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

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Textbook Question

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). 


b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2]. 

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Textbook Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

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Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=cos x, for 0≤x≤π/2; about the x-axis

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Textbook Question

"Determine whether the following statements are true and give an explanation or counterexample.


b. The volume of a hemisphere can be computed using the disk method. "

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