Calculus: Analytic Geometry and Calculus, with Vectors

10.2 Polar curves, tangents, and lengths 543

enable us to convert (7.26) into (10.281). For the special case in which
0(t) = t and ¢'(t) = 1, it is standard practice to put (10.281) in the form

(10.285) L = fa p jp2+

(Tdp^2

do

in which the variable of integration is (k

and//the

limits of integration are

called a and 0 (instead of a and b) because a and 0 "look more like angles."

It is worthwhile to know a little trick by which the above formulas

involving polar coordinates can be remembered. We can look at Figure

10.286 which shows, among other things, an arc of length As joining two

Figure 10.286

points P and Q which have polar coordinates p, 0 and p + Op, 95 + 04.

We can feel that the outer part of the figure resembles a rectangle enough

to enable us to write the approximate formulas

(10.287) As = p2 0¢2 + Opt, cos ¢ =

OP

P2 0O2 + Qp2

tan 4' =

and expect that correct results should be obtained by dividing by At

or by A¢ and taking limits. We can know that this optimism does not

prove formulas, but it can help us to recall theformulas when we have

forgotten them. When we wish to calculate the length L of the curve

having the polar equation p = f(4') with a < 0 < 0, we can, whenf is

continuous, sketch a figure more or less like Figure 10.286and use the

optimistic calculation

(10.288) L = lim As = lim J p2 XO2 -+ Apt

= lim p2 + fi() d¢

to lead us to the correct formula (10.285).