Calculus: Analytic Geometry and Calculus, with Vectors

4.7 Mass, linear density, and moments 259

as the definitionof the work done by F over the time interval a < t < b. Show
that (3) is equivalent to the definition

(4) W = f ba dt.

It remains for us to learn a little trick by means of which information
gleaned from this formula. Using the Newton formula F = ma gives

(5)

Hence

(6)

and

(7)

F(t)-v(t) =

m m dt jv(t)l2.

f r b

2 m Jbd

d

1v(1)12 I

1

a dt^1 m IX012]

L a

W = 2 mIv(b)I2 - 2 mlv(a)12.

can be

In case v(a) = 0, our work gives another derivation of the formula for the kinetic
energy of a particle of mass in having speed Iv(b)l.
14 The graph of the equation
a2x
Y = x2 + b2'

which usually appears in the disguised form x2y + b2y - a2x = 0, is called a
serpentine. Find the area (finite or infinite) of the region in the first quadrant
between the serpentine and its asymptote.
15 Accumulation of familiarity with Riemann sums may bring a desire to
b
know why fa f(x) dx cannot exist as a Riemann integral when f is defined but
unbounded over the interval a < x <- b. If the integral exists and has the value
I, then there must be a partition P of the interval a < x < b such that
n

(1) I , f(xk)Axk - II < 1

kml

whenever xk_1 S xk <- xk for each k. Show that if (1) holds, then

(2) lf(xl)I < (ix1)-1r1+ I + I f(xk)I AXk

L k=2

when xo <=xi S x1. This shows that f must be bounded over the first subinterval

of the partition P. Similar arguments show that f must be bounded over the

other subintervals and hence also over the whole interval a S x <_ b.

4.7 Mass, linear density, and moments This section involves

some ideas that turn out to be important in many ways. Let F be a

function which is defined over some finite interval a _< x < b and is