Calculus: Analytic Geometry and Calculus, with Vectors

(lu) #1
4.6 Riemann-Cauchy integrals and work 253

except when x = 0

fi


and cheerfully make the calculation

(4.637) i2 dx=-11 = -2 (????)
i x x 1-1

which would be correct if (4.636) were valid over the whole interval


-1 < x < 1. Since the wide world containsmany definitions of inte-

grals in addition to those of Riemann and Riemann-Cauchy, it is some-
what presumptuous to assert that (4.637) is ridiculous. However, when
we confine our attention to Riemann and Riemann-Cauchy integrals,
we can observe that (4.637) is incorrect.
Integrals of the types in (4.61) and (4.623) are particularly useful.
For example, the formula


(4.64)

1 fo. (x-M)2
e^202 dx = 1 (er > 0)

is not easily proved, but it lies at the foundation of very much work in
probability and statistics. Proof of this formula will appear later.
We conclude this section with a discussion of work in which the
formula

(4.65) (a > 0)


plays a fundamental role. To begin, we study the amount of work done


by a force F which pulls a particle P from the place on an x axis where
x = a to the place where x = b. The force F may have the direction of
the x axis but, as in Figure 4.651, this
is not necessarily so. Let f(x) denote F
the scalar component of the force F in
the direction of the motion, that is, in^0 a .X b x
the direction of the x axis. In case f (x) Figure 4.651
is a constant, measured in pounds (or
dynes), and the distance b - a is measured in feet (or centimeters),
the work W done by the force is measured in foot-pounds (or dyne-
centimeters) and is defined by the formula


(4.652) W = f(x)(b - a).


Since work and distance are scalars and force is a vector, it is quite
incorrect to perpetuate the ancient idea that "work is force times dis-


tance"; we must use scalar components of forces. In case the scalar
componentf(x) is different for different numbers x, the definition (4.652)
is inapplicable and we need integration to calculate W. The procedure
is almost identical with the procedure used to calculate areas and vol-
umes. We make a partition P of the interval from a to b with a "small"

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