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We already defined
work in one dimension

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is the product of force
times displacement

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for a constant force, but
now let's look at a case

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where we're applying a force,
f, that is a function of x.

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So, our component, f
of x, is a function

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of x in the i hat direction.

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And one of the simplest
examples of a force like this

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is what we call
the spring force.

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Now, when we apply this
force to our object--

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and let's look at an
object, i, i hat direction,

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we'll have an origin.

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And this is our plus-x
coordinate system.

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When we apply force--
what we want to look at

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is because the force is a
function of position-- then

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we want to look at
the displacement

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over a small amount.

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So let's call this the point xj.

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And out here, let's refer
to this as xj plus 1.

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And what we've
done is we're going

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to ask how much work is done
when the force is displaced

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from here to there.

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And that's what
we'll call delta x.

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So now our displacement is
a small displacement x of j

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plus 1 minus x of j.

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And our work for this small
displacement-- and that's

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why we'll indicate
it with a delta-- is

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equal to the force,
which is a function of x,

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times this displacement.

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And so we get f of x times
x of j plus 1 minus x of j.

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And what we have
to indicate here

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is because the force
is varying, we're

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looking at just this
displacement here.

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Let's refer to this force
as in the j-th part.

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The total work is just the sum
of all these scalar quantities.

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Remember, although force is
a vector and displacement

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is a vector, the product
of these two quantities

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is a scalar.

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And so, if we want
the total work,

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we have to sum from j goes
from 1 to n of this quantity

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f of j dot-- and I'll
put a little j there

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to indicate that-- delta xj.

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What does this sum mean?

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Well imagine that we're making
a series of displacements

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all the way out to
a final position,

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x final, and we divided
this interval into n pieces.

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And so this represents
the little bit

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of work done for all
of these displacements.

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And that is what we
define to be the work

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for a non-constant force.

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The issue here is about how
fine we cut this interval in.

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We made n individual pieces.

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But if we want to
ask ourselves, what

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is the limit as n
goes to infinity,

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then that's what we
now need to consider.

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So what we're doing
is we're making

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smaller and smaller and
smaller little displacements.

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And we're taking this
sum-- j goes from 1 to n

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of x of xj times delta xj.

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And this limit of a
sum is by definition

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the integral of the
force with respect to dx.

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So what we end up with is our
work is the integral of f of x.

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It's a function and I'm going to
have an integration variable, x

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prime of dx prime,
where x prime is

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going from our initial
position to our final position.

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And this is now our
definition of work

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which generalizes a constant
force to a non-constant force.

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Again, let's try to
look at some type

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of geometric interpretation.

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So if we plotted
f of x versus x.

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And now let's
consider a case where

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we have some arbitrary force.

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So I'm going to just draw the
force as if it were arbitrarily

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increasing as a function of x.

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And here is our x initial.

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And here's our x final.

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And again, we would like to
make a geometric interpretation.

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Let's consider xj.

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And I'll make this very big.

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xj plus 1 for the
sake of visualization.

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And this value here is f of xj.

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And so what we see now is
that little bit of work,

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like we had for
a constant force,

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can be approximated as the area
underneath the curve for just

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this small interval
between xj and xj plus 1.

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And if we make this
sum, what we're doing

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is we're just taking a
series of approximations

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to the area under these curves.

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Now you can see,
graphically, that there

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is very little error when the
function was nearly constant.

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But in this position, where
the function is growing,

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this represents
our little error.

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However, in the limit,
as n goes to infinity,

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we can make that error
go vanishingly small.

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So once again, we see, as
a geometric interpretation,

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that this is the area under
the curve of the force

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versus position
between the points

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initial xi and initial--
and the final, x

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final, where the
particle is being

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displaced from an
initial position

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to some final position.

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And here we can call our initial
position anywhere we want.

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If I write that, this x
initial, to the final position.

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And that's our generalization
of the definition of work.