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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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Integration Using Partial Fractions
Related Practice
Textbook Question

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.


b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).

Textbook Question

Power and energy The terms power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶ J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1W=1 J/s). Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6×10⁶ J. Suppose the power function of a large city over a 24-hr period is given by P(t) = E'(t) = 300 - 200 sin πt/12, where P is measured in megawatts and t=0 corresponds to 6:00 P.M. (see figure).


b. Burning 1 kg of coal produces about 450 kWh of energy. How many kilograms of coal are required to meet the energy needs of the city for 1 day? For 1 year? 

Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


b. Find the function that gives the amount of water in the tank at any time t≥0.

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

"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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Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


b. How far does the cyclist travel in the first 10 min?