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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 6.5.27a
Chapter 6, Problem 6.5.27a

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 = cos 2x, for 0 ≤ x ≤ π

Verified step by step guidance
1
Recall the formula for the arc length of a curve defined by 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 interval: here, \[y = \cos 2x\] with \[0 \leq x \leq \pi\]
Find the derivative of y with respect to x: \[\frac{dy}{dx} = \frac{d}{dx}(\cos 2x) = -2 \sin 2x\]
Square the derivative: \[\left(\frac{dy}{dx}\right)^2 = (-2 \sin 2x)^2 = 4 \sin^2 2x\]
Write the integral for the arc length using the formula: \[L = \int_{0}^{\pi} \sqrt{1 + 4 \sin^2 2x} \, dx\] This integral represents the arc length of the curve on the given interval.

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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 ∫ 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.
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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 = cos(2x). Differentiating y with respect to x gives dy/dx = -2 sin(2x), which is essential for substituting into the arc length integral.
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Derivatives of Other Trig Functions

Simplifying the Integral Expression

After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. Simplification may involve trigonometric identities to make the integral easier to evaluate, especially when using a calculator.
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Simplifying Trig Expressions
Related Practice
Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

a. On what intervals is the object moving in the positive direction?

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Textbook Question

Consider the following curves on the given intervals.  


a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis. 


y=tan x , for 0≤x≤π/4; about the x-axis 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.

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Textbook Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


a. Graph the velocity function for both riders. 

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Textbook Question

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

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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.

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