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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.75c
Chapter 8, Problem 8.2.75c

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...

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Recall that the average value of a function \(f(t)\) on an interval \([a, b]\) is given by the formula: \[\text{Average value} = \frac{1}{b - a} \int_a^b f(t) \, dt\]
Identify the function and the interval for this problem: Here, \(f(t) = s(t) = e^{-t} \sin t\), and the interval is \([n\pi, (n+1)\pi]\) where \(n = 0, 1, 2, \ldots\)
Set up the integral for the average value on the interval \([n\pi, (n+1)\pi]\): \[\text{Average value} = \frac{1}{(n+1)\pi - n\pi} \int_{n\pi}^{(n+1)\pi} e^{-t} \sin t \, dt = \frac{1}{\pi} \int_{n\pi}^{(n+1)\pi} e^{-t} \sin t \, dt\]
To evaluate the integral \(\int e^{-t} \sin t \, dt\), use integration by parts or recognize it as a standard integral involving exponential and trigonometric functions. The integral has a known form: \[\int e^{at} \sin(bt) \, dt = \frac{e^{at}}{a^2 + b^2} (a \sin(bt) - b \cos(bt)) + C\] In this problem, \(a = -1\) and \(b = 1\).
Apply the definite integral limits \(t = n\pi\) and \(t = (n+1)\pi\) to the antiderivative found in the previous step, then substitute back into the average value formula to express the average value in terms of \(n\). This will give a general formula for the average position on the interval \([n\pi, (n+1)\pi]\).

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Key Concepts

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

Average Value of a Function

The average value of a function f(t) over an interval [a, b] is given by (1/(b - a)) times the integral of f(t) from a to b. It represents the mean height of the function on that interval and is useful for understanding the overall behavior of oscillating or varying functions.
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Average Value of a Function

Integration of Exponential and Trigonometric Functions

Integrating functions like e^(-t) sin(t) involves techniques such as integration by parts or recognizing standard integral forms. Understanding how to handle the product of an exponential decay and a sinusoidal function is essential for solving problems involving damped oscillations.
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Integrals of General Exponential Functions

Properties of Oscillatory Functions on Intervals of Length π

Sine functions have a period of 2π, but analyzing them over intervals of length π, such as [nπ, (n+1)π], helps in studying half-period behavior. This is important when combined with damping factors like e^(-t), as it affects the average value and sign of the oscillations over each interval.
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Properties of Functions
Related Practice
Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

46. ∫(0 to 2) x⁴ dx; n = 30

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

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Textbook Question

2. Give an example of each of the following.

b. A repeated linear factor

Textbook Question

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

Textbook Question

59. Two Methods

b. Evaluate ∫(x / √(x + 1)) dx using substitution.

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Textbook Question

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.