Calculus: Analytic Geometry and Calculus, with Vectors

548 Polar, cylindrical, and spherical coordinates

in which the subscripts and star superscripts do not appear. In applica-
tions, we often write p or p(o) in place of f(o).
To add variety to our acquaintance with problems, we define two
numbers II and 12 by the formulas

(10.34) I1 = f to [p(t)J2,p'(t) dt, IZ =^1 f ,_ [p(t)121 0'(t) I dt

and ask how II and 12 may be interpreted in terms of polar coordinates

and areas when t1 < t2. In order that the integrands and integrals exist,

it is necessary that p and 0 be functions for which p(t), 0(t), and 0'(t)

exist when t1 < 1 < t2, and we simplify matters by supposing that p,

¢, and 0' are all continuous over t1 5 t < t2. As t increases over the

interval t1 <_ t 5 t2, the point P(t) having polar coordinates (p(t), 0(t))

traces a curve (or an oriented curve) C from P(t1) to P(t2) that could look

like that shown in Figure 10.35 or like that shown in Figure 10.36.

r(t1) = r(t2)

Figure 10.35

A I

Figure 10.36

Of course, there are other possibilities, and we exclude more elaborate

ones by supposing that the interval ti _< t t2 can be separated into a

finite set of subintervals such that, over each subinterval, 0 is either

monotone increasing or monotone decreasing. Let P be a partition of

the interval t1 5 t__<_t2 such that, in each subinterval, 0(t) is either

monotone increasing or monotone decreasing. Let tk* be chosen in the

kth subinterval in such a way that

(10.37) b(tk) - 0(tk-1) = O'(tk)(tk - tk-1) = 0'(tk) Otk,

and build the Riemann sum

(10.38) Atk

which approximates I. A particular term in this sum is an approxima-

tion to the area of the region swept over by the vector from 0 to P(t)

as t increases from tk_1 to tk provided 0(tk) > t(tk_1), that is, provided

the vector rotates in the positive direction. Similarly, the term is an

approximation to the negative of the area if the vector rotates in the

negative direction. Thus II is the sum of areas of regions swept over

in the positive direction and the negative of areas of regions swept over