   Chapter 8.2, Problem 91E

Chapter
Section
Textbook Problem

Average Displacement A damping force affects the vibration of a spring so that the displacement of the spring is given by y = e − 4 t ( cos   2 t + 5 sin   2 t ) . Find the average value of y on the interval from t = 0 to t = π .

To determine

To calculate: We will calculate the average value of y on the interval from t=0 to t=π where y is given as:

y=e4t(cos2t+5sin2t).

Explanation

Given:

A damping force affects the vibration of a spring so that the displacement of the spring is given by

y=e4t(cos2t+5sin2t)

Interval is t=0 to t=π.

Formula used:

Average value 1baabf(x)dx and Integration by parts.

Calculation:

Consider the following displacement of a spring,

y=e4t(cos2t+5sin2t)

The average value of y=f(x) in the interval (a,b) is;

Average value 1baabf(x)dx.

Hence, the average value of y=e4t(cos2t+5sin2t) in the interval (0,π).

We obtain that the average value is 1π0πe4t(cos2t+5sin2t)dt …… (1)

Now, apply integration by parts formula, to integrate the integrals e4tcos2tdt and e4tsin2tdt

First, integrate the integral e4tcos2tdt as follows:

Apply integration by parts using formula udv=uvvdu, to get,

I=e4tcos2tdt=cos2t(e4t4)(e4t4)(2sin2tdt)=e4t4cos2t12e4tsin2tdt …… (2)

Now, again use integration by parts formula udv=uvvdu to integrate the integral e4tsin2tdt.

I=e4tsin2tdt=sin2t(e4t4)(e4t4)(2cos2tdt)=e4t4sin2t+12e4tcos2tdt

Now, substitute the above integral value in equation (2), to get:

I=e4tcos2tdt=e4t4cos2t12(e4t4sin2t+12e4tcos2tdt)=14e4tcos2t+18e4tsin2t14e4tcos2tdt+c=14e4tcos2t+18e4tsin2t14I+cI+I4=14e4tcos2t+18e4tsin2t+c

Simplifying further,

54I=14e4tcos2t+18e4tsin2t+cI=45(14e4tcos2t+18e4tsin2t+c)=15e4tcos2t+110e4tsin2t+C

Thus, the integral value of e4tcos2tdt will be;

e4tcos2tdt=15e4tcos2t+110e4tsin2t+C

We willintegrate the integral e4tsin2tdt as follows:

Let I=e4tsin2tdt

Apply integration by parts formula udv=uvvdu:

I=e4tsin2tdt=sin2t(e4t4)(e4t4)(2cos2tdt)=e4t4sin2t+12e4tcos2tdt …… (3)

Now again applyintegration by parts formula

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