INTERNATIONAL EDITION---Numerical Methods for Engineers, 7th edition
INTERNATIONAL EDITION---Numerical Methods for Engineers, 7th edition
7th Edition
ISBN: 9781259170546
Author: Steven Chapra and Raymond Canale
Publisher: MCG
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Textbook Question
Chapter 19, Problem 1P

The average values of a function can be determined by

f ( x ) ¯ = 0 x f ( x ) d x x

Use this relationship to verify the results of Eq. (19.13).

Expert Solution & Answer
Check Mark
To determine

To prove: The following results,

sin(ω0t)N=0,cos(ω0t)N=0,sin2(ω0t)N=12,cos2(ω0t)N=12,cos(ω0t)sin(ω0t)N=0

If the average value formula is f(x)¯=0xf(x)dxx.

Explanation of Solution

Given:

The average value formula is f(x)¯=0xf(x)dxx and the results are,

sin(ω0t)N=0,cos(ω0t)N=0,sin2(ω0t)N=12,cos2(ω0t)N=12,cos(ω0t)sin(ω0t)N=0

Formula used:

The integral formula are abf(x)dx=f(b)f(a),sin(nx)dx=cos(nx)n,cosxdx=sinx.

The trigonometric identity are,

sin2x=1cos(2x)2,cos2x=1+cos(2x)2,sin(2x)=2sin(x)cos(x)

Proof:

Consider the identity,

sin(ω0t)N

Use average value formula and take f(x)=sinω0t and T=N.

As it is known that the angular frequency ω0 is the fraction of 2π and time period T as shown below:

Therefore,

sin(ω0t)N=0Tsin(ω0t)dtT

Use integration formula,

0Tsin(ω0t)dtT=(1ω0T)[cos(ω0t)]0T=(12πTT)[cos(2πTt)]0T=(12π)[cos(2π)cos(0)]=(12π)[11]

On simplifying further,

sin(ω0t)N=0

Hence, the result is sin(ω0t)N=0.

Now consider next result,

cos(ω0t)N

Use average value formula and take f(x)=cosω0t and T=N.

As it is known that the angular frequency ω0 is the fraction of 2π and time period T as shown below:

Therefore,

cos(ω0t)N=0Tcos(ω0t)dtT

Use the integral formula,

0Tcos(ω0t)dtT=(1ω0T)[sin(ω0t)]0T=(12πTT)[sin(2π)sin(0)]=(12πTT)[00]=0

Hence, the result is cos(ω0t)N=0.

Now, consider next result,

sin2(ω0t)N

Use average value formula and take f(x)=sin2ω0t and T=N.

As it is known that the angular frequency ω0 is the fraction of 2π and time period T as shown below:

Therefore,

sin2(ω0t)N=0Tsin2(ω0t)dtT

Use the trigonometric identity,

0Tsin2(ω0t)dtT=(12T)0T(1cos(2ω0t))dt

Use the integral formula,

0Tsin2(ω0t)dtT=(12T)[tsin(2ω0t)2ω0]0T=(12T)[Tsin(2(2πT)T)2(2πT)0+sin(0)]=(12T)[T00+0]=12

Hence, the result is sin2(ω0t)N=12.

Now, consider next result,

cos2(ω0t)N

Use average value formula and take f(x)=cos2ω0t and T=N.

As it is known that the angular frequency ω0 is the fraction of 2π and time period T as shown below:

Therefore,

cos2(ω0t)N=0Tcos2(ω0t)dtT

Use the trigonometric identity,

0Tcos2(ω0t)dtT=(12T)0T(1+cos(2ω0t))dt

Use the integral formula,

0Tcos2(ω0t)dtT=(12T)[t+sin(2ω0t)2ω0]0T=(12T)[t+sin(2(2πT)T)2(2πT)0sin(0)]0T=(12T)[T+sin(4π)2(2πT)00]=(12T)[T+000]0T

On further simplifying,

0Tcos2(ω0t)dtT=12

Hence, the result is cos2(ω0t)N=12.

Now, consider next result,

cos(ω0t)sin(ω0t)N

Use average value formula and take f(x)=(cosω0t)(sinω0t) and T=N.

As it is known that the angular frequency ω0 is the fraction of 2π and time period T as shown below:

Therefore,

cos(ω0t)sin(ω0t)N=0Tcos(ω0t)sin(ω0t)dtT

Use the trigonometric identity,

cos(ω0t)sin(ω0t)N=(12T)0Tsin(2ω0t)dt=(14Tω0)[cos(2ω0t)]0T=(14T(2πT))[cos(2(2πT)T)cos(0)]0T=(14T(2πT))[cos(4π)cos(0)]

On further simplifying,

cos(ω0t)sin(ω0t)N=(14T(2πT))[11]=0

Hence, the result is cos(ω0t)sin(ω0t)N=0.

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