Calculus: Analytic Geometry and Calculus, with Vectors

426 Curves, lengths, and curvatures

points all travel approximately the same distance 2axk and that the segment
generates a surface essentially like a ribbon having width Lk and length 27rxk.
This leads us to the intuitive idea that one term in the sum

(2) ISI = lim 2;2ir 1 [[f'(xk)]2 xk t.xk

should, when the norm fPf of P is small, be a good approximation to the area of the
part of S generated by one segment of the graph G. The next step is to adopt the
tentative intuitive conclusion that the sum is a good approximation to the total
area JSI of S when the norm of P is small or, in other words, that (2) should be
valid. Our theory of Riemann sums and integrals assures us that if (2) is true,
then

(3) BSI = 21r fab 1 + [f'(x)]2 x dx.

If we have sufficient confidence in our calculations (it would, of course, be fatal
to use an incorrect formula for the circumference of a circle of radius xk) and in
our intuitive ideas, we can install the formula (3) as one (not the) definition of
area of surfaces of revolution. It does not make sense to claim that this definition
is "correct," but experience shows that it is useful and this is all that we can
expect from definitions. For the case in which f(x) = kx2, a = 0, and b = r,
(3) reduces to

ISI = 2a for (1 + 4k2x2)' x dx =6k2[(1 + 4k2r2)h 1].

When k and r = 6, this reduces to BSI = 491r.

15 We invest a little time to look at some curve integrals that are called line

integrals by those who adhere to the notion

that curves are lines. Let functions x(t), y(t),

z(t), the point P(t), the vector r(t), the curve

C, the partition P of a 5 t < b, and the num-

bers Otk, /xk, ayk, Ozk be defined as in the part

of this section preceding (7.231). Let a vector

function F having scalar components fI, f2, fa

be defined and continuous over a part of Es

y

z that contains the curve C. We consider F to

Figure 7.294 be a force which operates upon a particle at

P(t) as the particle moves from P(a) to P(b).

The schematic Figure 7.294 may be helpful. With the idea that the work done

by F as the particle moves from P(tk_I) to P(tk) is closely approximated by

(1)

when the norm of the partition P is small, we can define the total work W done

by F by the formula

(2) W = lim 2;F(xk,yk,zk)I.rk-