Find the work done by the force field F in moving an object from P(-5, 9) to Q(3, 5).

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crmhaske | College Teacher | (Level 3) Associate Educator

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First we must parametrize the path:

r(t) = (-5,9) + t(8,-4) =(-5+8t,9-4t) for t [0,1]

Now, in order to solve for work:

W = `int_rF*dr=int_0^1((-10+16t)/(9-4t),(-5+8t)^2/(9-4t)^2)*(8,-4)dt`

` `

Solve the dot product `F*R:`

W=`int_0^1((-80+128t)/(9-4t)-4(-5+8t)^2/(9-4t)^2)dt`

Simplify:

W=`int_0^1((-720+1472t-512t^2-100+320t-256t^2)/(9-4t)^2)dt`

`W=int_0^1((-820+1792t-768t^2)/(9-4t)^2)dt`

This integral can be solved using integration by partial fractions:

W=`int_0^1(-48-(416/(4t-9))-(676/(4t-9)^2))dt`

`W=int_0^1(-48dt)-int_0^1(416/(4t-9))dt-int_0^1(676/(4t-9)^2)dt`

The first integral is solved using simple integration:

-48(1-0) = -48

 

The second integral is solved using integration by substitution:

z = 4t-9 and dz = 4dt therefore dt = dz/4 

`int_0^1(416/z)(dz/4)=int_0^1(104/z)dz=[104log(z)]_0^1`

`[104log(z)]_0^1=[104log(4t-9)]_0^1`

 

The last integral is also solved using integration by substitution:

z = 4t - 9 and dz = 4 dt therefore dt = dz/4

`int_0^1(676/(4t-9))dt = int_0^1(676/z^2)(dz/4) = int_0^1(169/z^2)dz`

`=[-169/z]_0^1=[-169/(4t-9)]_0^1`

`=-169/-5-(-169/-9)=33.8-18.8=15`

Therefore the solution is:

W = -48-(-25.58)-15 = -37.4

Sources:

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