Calculus: Analytic Geometry and Calculus, with Vectors

264 Integral$

unit vector in its direction. Our next step is to write PPk in terms of the coordi.
nates of P and P, and to write
(
(4) [1Fk = Gm E S (tk)

tk - x)i - yj - xk

7

Otk.

[(lk - x)2 + Y2 + zJ

Everything is now prepared for the crucial steps. When the norm JQJ of the

partition Q is small, the sums in (4) should be good approximations for the force

F that we are trying to define. In other words, F should be the limit of these

sums. But these sums are Riemann sums and, provided P(x,y,z) is not a point

on the interval a --<- x S b of the x axis, they have a limit which is the Riemann

integral in the formula

(5) F = Gmrb (t - x)i - yj - zk

Ja a(t) ((t- x)2 + Y2 + z2]3

dt.

Our work motivates the definition whereby F is defined by (5). While (5) serves

as a source of information about F in other cases, we confine our attention here

to the case in which the density is a constant, say &(t) = bo for each t, and, more-

over, y = z = 0 and x < a < b. In this case, F has the direction of i, and if

we denote its magnitude by Fi(a,b,x), then

(6)

It is easy to see that

b

Fi(a,b,x) = Gmbo

f.

a (t - x)-2 dt

• 1
  = Gmbo

(1

a-x b-x

(7) lim Fi(a,b,x) =Gmbo lim Fi(a,b,x) = eo.

b--.m a - x x-.a-

If these formulas agree with our intuitive notions, then at least some of our

intuitive notions are good. The second result in (7) gives us a lesson in approxi-

mation. Since particles near ends of actual steel rods are not subject to huge

attractive forces, we must conclude that very bad approximations to forces on

these particles are obtained from calculations based on assumptions that the

rods are concentrated on their axes.

(^11) Modify Figure 4.792 to fit the case in which h > 0, a = -h, b = h,
b(x) = bo, x = 0, and z = 0. Show that, in this case, formula (5) of the pre-
ceding problem becomes
ti-
F =Gmbo f
h Yj
-h (t' {- y2),,dt.
After having a good look at the coefficients of i and j, show that
(h
F = -2GmboyjJo (t2+
Y2)
dt.
12 A thin cylindrical shell S of radius R has its axis on the x axis of an x, y,
z coordinate system and has its ends in the planes having the equations x = a
and x = b. This shell has constant areal density (mass per unit area) S. Find
the gravitational force F which it exerts upon a particle m* ofmass in which is