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212 CHAPTER 7 Work and Energy

A contribution to the work: Note that each of the terms F (x ) x in the sum is the area of a rectangle of height F (x )
x
i
i
x
the product F x , which and width x, highlighted in color in Fig. 7.11. Thus, Eq. (7.13) gives the sum of all

x
is this rectangle’s area.
F x the rectangular areas shown in Fig. 7.11.
Equation (7.13) is only an approximation for the work. In order to improve this
approximation, we must use a smaller interval x. In the limiting case x S 0 (and
n S ), the width of each rectangle approaches zero and the number of rectangles
approaches infinity, so we obtain an exact expression for the work. Thus, the exact
F x
definition for the work done by a variable force is
n
lim
W ¢xS0 a F (x ) ¢x

i
x
x i 1
a x i–1 x i b
This expression is called the integral of the function F (x) between the limits a and
x
x b. The usual notation for this integral is
x is the width b
of each interval. W F (x) dx (7.14)
x
a
FIGURE 7.11 The curved plot of F x
vs. x has been approximated by a series of where the symbol is called the integral sign and the function F (x) is called the inte-
x
horizontal and vertical steps. This is a good grand. The quantity (7.14) is equal to the area bounded by the curve representing F (x),
approximation if x is very small. x
the x axis, and the vertical lines x a and x b in Fig. 7.12. More generally, for a curve
that has some portions above the x axis and some portions below, the quantity (7.14)
is the net area bounded by the curve above and below the x axis, with areas above the
x axis being reckoned as positive and areas below the x axis as negative.
We will also need to consider arbitrarily small contributions to the work. From
Eq. (7.12), the infinitesimal work dW done by the force F (x) when acting over an
x
infinitesimal displacement dx is
dW F (x) dx (7.15)
x
We will see later that the form (7.15) is useful for calculations of particular quantities,
such as power or torque.
Finally, if the force is variable and the motion is in more than one dimension, the
work can be obtained by generalizing Eq. (7.7):

#
work done by a variable force W F ds (7.16)

To evaluate Eq. (7.16), it is often easiest to express the integral as the sum of three
This area is work done
by F during motion integrals, similar to the form of Eq. (7.9). For now, we consider the use of Eq. (7.14)
x
F x from x = a to x = b. to determine the total work done by a variable force as it acts over some distance in
one dimension.

A spring exerts a restoring force F (x) kx on a particle
EXAMPLE 4 x
attached to it (compare Section 6.2).What is the work done by
the spring on the particle when it moves from x a to x b?

x SOLUTION: By Eq. (7.14), the work is the integral
a b
FIGURE 7.12 The integral b F (x)dx is
a x b b
the area (colored) under the curve represent- W F (x) dx ( kx) dx
x
ing F (x) between x a and x b. a a
x