Question

Average valueIn a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t isy = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.Find the average value of y over the interval 0 ≤ t ≤ 2π.

Accepted Answer

Recall that the average value of a function $$y = f(t)$$ over the interval $$[a, b] is $$given by the formula:

$$	ext{Average value} = \frac{1}{b - a} \int_a^b f(t) \, dt

In $$this problem, $$a = 0$$ and $$b = 2\pi. $$Substitute the given function $$y = 4e$$^{-t}$$(\sin(t) - \cos(t))$$ into the average value formula:

$$	ext{Average value} = \frac{1}{2\pi - 0} \int_0^{2\pi} 4e^{-t}(\sin(t) - \cos(t)) \, dt = \frac{1}{2\pi} \int_0^{2\pi} 4e^{-t}(\sin(t) - \cos(t)) \, dt$$ Factor out the constant 4 from the integral to simplify:

$$	ext{Average value} = \frac{4}{2\pi} \int_0^{2\pi} e^{-t}(\sin(t) - \cos(t)) \, dt = \frac{2}{\pi} \int_0^{2\pi} e^{-t}(\sin(t) - \cos(t)) \, dt$$ Split the integral into two separate integrals to handle each term individually:

$$\int_0^{2\pi} e^{-t}(\sin(t) - \cos(t)) \, dt = \int_0^{2\pi} e^{-t} \sin(t) \, dt - \int_0^{2\pi} e^{-t} \cos(t) \, dt$$ Use integration by parts or recall the standard integrals for $$\int e$$^{at}$$ \sin(bt) \, dt$$ and $$\int e$$^{at}$$ \cos(bt) \, dt to $$evaluate each integral. Then substitute the evaluated integrals back into the expression for the average value.

Average value
In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is
y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.
Find the average value of y over the interval 0 ≤ t ≤ 2π.

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Concetti chiave

Average Value of a Function

Definite Integration

Exponential and Trigonometric Functions