The force applied against a moving object travelling on a linear path is given by #F(x)=3xe^x #. How much work would it take to move the object over #x in [2, 4] #?

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Answer 1 · 2016-07-01T22:26:29

#= 9e^4 - 3e^2#

In the simplest sense, Work = Force X Distance

[More specifically: #W = int_C vecF(vec r) * d vec r#]

Here, in 1 dimension, #x#, we can say that

#W = int dx qquad F(x)#

So

#W = int_2^4 dx qquad 3xe^x#

this is integration by parts ie #int uv' = uv - int u'v#

so

#u = 3x, u' = 3#
#v' = e^x, v = e^x#

so we have

#W = [3x e^x]_2^4 - int dx qquad 3 e^x#

#= [3x e^x]_2^4 - [3 e^x]_2^4#

# = 3[e^x (x-1) ]_2^4 #

# = 3 {[e^4(4-1) ] - [e^2(2-1) ] }#

#= 9e^4 - 3e^2#