Question

Average Value A retarding force, symbolized by the dashpot in the figure, slows the motion of the weighted spring so that the mass's position at time \(t\) is $$y=2 e^{-t} \cos t, \quad t \geq 0$$ Find the average value of \(y\) over the interval \(0 \leq t \leq 2 \pi\)

Short answer

The average value of \(y = 2e^{-t}\cos t\) over the interval \(0 \leq t \leq 2\pi\) is 0.

Step-by-step solution

Identify the Function and Interval

We first identify the function and the interval. The function is \(y = 2e^{-t}\cos t\) and the required interval over which the average value needs to be found is \(t = 0\) to \(t = 2\pi\).

Define Average Function Value Formula

Next, we recall that the formula for finding the average value of a function \(f(x)\) over the interval [a, b] is given by \(1/(b-a)\int_a^bf(x) dx\). We substitute \(f(x) = 2e^{-t}\cos t\), \(a = 0\) and \(b = 2\pi\) into the formula.

Evaluate the Integral

To solve the integral \(\int_0^{2\pi}2e^{-t}\cos t dt\), it's necessary to use integration by parts, where the rule states if an integral is in the form \(uv'\), its integral is \(uv - \int vu' dx\). Let \(u = e^{-t}\) and \(v' = 2\cos t\), after some steps of simplifications, we find that the integral equals \(e^{-2 \pi} sin(2 \pi)\).

Substitute and Solve

Using the limits of the integral, solve \(1/(2\pi - 0)\times [e^{-2 \pi} sin(2 \pi) - e^{-0} sin(0)]\), which equals to 0.