Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by $y = e ^ { - 8 t } ( \cos 2 t + 9 \sin 2 t )$. Find the average value of $y$ on the interval from $t = 0$ to $t = \pi$. Round your answer to three decimal places.
A) $0.406$
B) $0.131$
C) $0.122$
D) $0.129$
E) $0.382$

Question

Quizplus · Accepted Answer

The average value of a function $y(t)$ over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} y(t) \, dt$. Here, $y(t) = e^{-8t} (\cos 2t + 9 \sin 2t)$ and the interval is $[0, \pi]$. Evaluating the integral and dividing by $\pi$, the average value is approximately $0.122$.

Suppose a Damping Force Affects the Vibration of a Spring y=e−8t(cos⁡2t+9sin⁡2t)y = e ^ { - 8 t } ( \cos 2 t + 9 \sin 2 t )y=e−8t(cos2t+9sin2t)

Suppose a Damping Force Affects the Vibration of a Spring $y = e ^ { - 8 t } ( \cos 2 t + 9 \sin 2 t )$