Chapter 7.3, Problem 35E

To determine

To find the average value of the mass of the spring $y=2e^{-t}\cos t$ at time $t\ge 0$

Answer to Problem 35E

The average value of the mass of the spring is $\frac{1-e^{-2\pi }}{2\pi }$

Explanation of Solution

Given information: The position of the mass of the spring is $y=2e^{-t}\cos t$ at time $t\ge 0$

Formula used: The average value of a function $f(x)$ on $[a,b]$ is $\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$

  ${\displaystyle \int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}\left[a\cos bx+b\sin bx\right]+C}$

Calculation :

For $0\le t\le 2\pi$ ,the average value = $\frac{1}{2\pi -0}{\displaystyle \underset{0}{\overset{2\pi }{\int }}2e^{-t}\cos tdt}$

  $=\frac{2}{2\pi }{\left(\frac{e^{-t}}{1^2+1^2}\left[-\cos t+\sin t\right]\right)}_0^{2\pi }$

  $=\frac{1}{2\pi }\left(e^{-2\pi }(-1)+1\right)=\frac{1-e^{-2\pi }}{2\pi }$

Hence, the average value of the mass of the spring is $\frac{1-e^{-2\pi }}{2\pi }$