Textbook Question
49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:
c. ∫ v du = u·v - ∫ u dv
1
views
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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, ...
Verified step by step guidance1
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 \([n\pi, (n+1)\pi]\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mKey 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.
Recommended video:
Guided course
06:37
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.
Recommended video:
05:11Integrals 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.
Recommended video:
Guided course
06:21Properties 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?
1
views
Textbook Question
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0
Textbook Question
60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.
Textbook Question
94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],
where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.
c. Verify by integration that the position function is given by
s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],
where s'(t) = v(t) and s(0) = 0.
1
views