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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.R.109
Capitolo 8, Problema 8.R.109

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by
v(t) = 10 * sin(3t).

Guida verificata passo dopo passo
1
Recall that the average velocity over the interval \([a, b]\) is given by the formula: \(\text{Average velocity} = \frac{1}{b - a} \int_a^b v(t) \, dt\).
Identify the interval and the velocity function: here, \(a = 0\), \(b = \pi\), and \(v(t) = 10 \sin(3t)\).
Set up the integral for the average velocity: \(\frac{1}{\pi - 0} \int_0^{\pi} 10 \sin(3t) \, dt = \frac{1}{\pi} \int_0^{\pi} 10 \sin(3t) \, dt\).
Evaluate the integral \(\int_0^{\pi} 10 \sin(3t) \, dt\) by using the substitution method or by recalling the integral of sine: \(\int \sin(k t) \, dt = -\frac{1}{k} \cos(k t) + C\).
After finding the definite integral value, divide it by \(\pi\) to find the average velocity over the interval \([0, \pi]\).

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Concetti chiave

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Average Velocity

Average velocity over a time interval is defined as the total displacement divided by the total time elapsed. It can be calculated by integrating the velocity function over the interval to find displacement, then dividing by the length of the interval.
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Average Value of a Function

Definite Integral of Velocity

The definite integral of a velocity function v(t) from time a to b gives the net displacement of the object during that interval. This integral is essential to find total change in position when velocity varies with time.
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Definition of the Definite Integral

Trigonometric Integration

Integrating functions involving sine or cosine requires knowledge of basic trigonometric integrals. For example, the integral of sin(k t) with respect to t is -cos(k t)/k, which is used to evaluate the displacement from the given velocity function.
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Introduction to Trigonometric Functions
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Domanda del libro di testo

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

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82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

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96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

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