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 guidance

1. Recall that the average value of a function \(f(t)\) on an interval \([a, b]\) is given by the formula: \[\text{Average value} = \frac{1}{b - a} \int_a^b f(t) \, dt\]
2. 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 \(n = 0, 1, 2, \ldots\)
3. Set up the integral for the average value on the interval \([n\pi, (n+1)\pi]\): \[\text{Average value} = \frac{1}{(n+1)\pi - n\pi} \int_{n\pi}^{(n+1)\pi} e^{-t} \sin t \, dt = \frac{1}{\pi} \int_{n\pi}^{(n+1)\pi} e^{-t} \sin t \, dt\]
4. 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 \(b = 1\).
5. 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 \([n\pi, (n+1)\pi]\).

Key Concepts

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.

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.

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.