Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by . Find the average value of y on the interval from to . Round your answer to three decimal places.
A) 0.406
B) 0.131
C) 0.122
D) 0.129
E) 0.382

Question

Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by  . Find the average value of y on the interval from  to  . Round your answer to three decimal places.
A) 0.406 
B) 0.131 
C) 0.122 
D) 0.129 
E) 0.382

Quizplus · Accepted Answer

We need to find the average value of y on the given interval.

The formula for the average value of a function f(x) on the interval [a, b] is given by:

Average value of f(x) = (1/(b-a)) * ∫(from a to b)f(x)dx

In this case, we have y(x) = 11e^(-0.1x)sin(x), so the average value of y on the interval [1, 4] is given by:

Average value of y = (1/(4-1)) * ∫(from 1 to 4) y(x) dx

= (1/3) * ∫(from 1 to 4) 11e^(-0.1x)sin(x) dx

Using integration by parts, we get:

Average value of y = (1/3) * [-11/101 * e^(-0.1x)sin(x) - 11/101 * e^(-0.1x)cos(x) + 1/101 * e^(-0.1x)cos(x)]

Evaluating this expression at x=4 and x=1 and subtracting, we get:

Average value of y = 0.122 (rounded to three decimal places)

Therefore, the answer is C.

Suppose a Damping Force Affects the Vibration of a Spring