Calculus: Analytic Geometry and Calculus, with Vectors

550 Polar, cylindrical, and spherical coordinates

9 It is much easier to learn to play a violin than to acquire competence to

give basic definitions and theorems involving areas of patches of curved surfaces.

Some problems are so simple, however, that elementary methods yield answers

that are universally considered to be correct. As Figure 10.391 suggests, a

Figure 10.391

hemisphere (or hemispherical surface) of radius a is generated by rotating a

quadrant of radius a about the y axis. To find the area .4 of this hemisphere,

we make a partition of the interval 0 5 0< 7r/2. Rotating the segment of

length a A,i about the y axis gives a part of the hemisphere that can be roughly

described as a ribbon of width a A¢ and length 2ar, where r is a cos 0 The

process for setting up and evaluating integrals then gives

14= lim 12,ra2 cos q5 A0 = 2ira2

Iox'2

cos0 d¢= 2ira2.

These preliminaries can be ended with the remark that the area of the hemi-

sphere ought to be about double the area of the equatorial circular disk and that

the world is so simple that the factor is exactly 2. Now comes the problem.

Supposing that 0 < c < c + h <= a, find the area of the zone generated by

rotating (about the y axis) the part of the arc between the lines having the equa-

tions y = c and y = c + h.

10 Supposing that r(t) and 0(t) have continuous derivatives and that

x(t) = r(t) cos 0(t), y(t) = r(t) sin 4(t),

calculate x'(t) and y'(t) and show that

x(t)y'(t) - y(t)z(t) {

11 Making use of the method involving (10.38), put appropriate hypotheses

on functions x(t) and y(t) and discover geometric interpretations of the integrals

r`:

x(t)y'(t) dt,

t2

y(t)x'(t) dt.

12 Show that if a particle P moves on the conic having the polar coordinate

equation

(1) P(t) =1 - c os q5(t)