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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 8.2.75a
Chapter 8, Problem 8.2.75a

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
a. Graph the position function. At what times does the oscillator pass through the position s = 0?

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1
Identify the position function given: \(s(t) = e^{-t} \sin t\). This function describes the displacement of the mass on the spring over time.
To find when the oscillator passes through the position \(s = 0\), set the position function equal to zero: \(e^{-t} \sin t = 0\).
Since \(e^{-t}\) is never zero for any real \(t\), the zeros of \(s(t)\) occur when \(\sin t = 0\).
Recall that \(\sin t = 0\) at integer multiples of \(\pi\), so the times when the oscillator passes through \(s=0\) are \(t = n\pi\), where \(n\) is any integer (0, 1, 2, 3, ...).
To graph the position function, plot \(s(t) = e^{-t} \sin t\) over a suitable interval (for example, \(t \geq 0\)), noting that the amplitude of the oscillations decreases exponentially due to the \(e^{-t}\) factor, and the zeros occur at multiples of \(\pi\).

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Key Concepts

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

Damped Harmonic Motion

Damped harmonic motion describes oscillations where the amplitude decreases over time due to friction or resistance. In the given function s(t) = e⁻ᵗ sin t, the exponential term e⁻ᵗ represents the damping effect, causing the oscillations to gradually reduce in magnitude as time increases.
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P-Series and Harmonic Series

Zeros of a Function

Finding when the oscillator passes through position s = 0 involves solving s(t) = 0. Since s(t) is a product of e⁻ᵗ and sin t, and e⁻ᵗ is never zero, the zeros occur when sin t = 0. This happens at integer multiples of π, i.e., t = nπ for n = 0, 1, 2, ...
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Graphing Transcendental Functions

Graphing s(t) = e⁻ᵗ sin t requires understanding how the sinusoidal oscillations are modulated by the exponential decay. The graph shows oscillations with peaks decreasing exponentially, illustrating the combined effect of sine wave behavior and damping over time.
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Related Practice
Textbook Question

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].

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Textbook Question

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

a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

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Textbook Question

A piece of wood paneling must be cut in the shape shown in the figure.

The coordinates of several points on its curved surface are also shown (with units of inches).

a. Estimate the surface area of the paneling using the Trapezoid Rule.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

69. Let f(x) = sin(eˣ).

a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

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Textbook Question

42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.

a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule

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Textbook Question

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.

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