Calculus: Analytic Geometry and Calculus, with Vectors

13.5 Integrals in polar coordinates 689

Examplel Let05a<#5,r,letf(4)>_Owhen a<¢_<S,letf

be Riemann integrable over the interval a < 0 < #, and let S be the

plane set of points having polar coordinates (p,4)) for whicha < 0 < ft

and 0 < p < f(4)). In case f is continuous, S is the set bounded by the

polar graphs of the equations 0 = a, ¢ = S, and p = f(o). The problem

is to find the volume JBI of the solid body B that is "generated" by rotat-

ing the set S about the initial line (or x axis) of Figure 13.56. The first

Figure 13.56

step is to partition S into subsets by radial lines through the origin and
circular arcs having centers at the origin. Letting p and .0 denote the
polar coordinates of a point in a typical subset, we use the number p £p 04,
as an approximation to the area of the subset. When the subset is
rotated about the x axis, it generates a solid which may be thought of as a
ring or hoop or gasket having radius p sin 0 and length 2rp sin 0. The
number

or

2,rp2 sin 0 Op A4),

being the product of the length of the ring and the area of a cross section
of the ring, is then used as an approximation to the volume of the ring.
The sum in

(2rpsin0) (p Op o4)),

x

(13.561) CBI = lim E2rp2 sin 4) Lp 04)

is then used as an approximation to the volume IBI of the body B, and

the limit is (without proof) taken to be the exact volume jBI. The

right member of (13.561) is a double integral. Expressing this as an

iterated integral gives

(13.562) (BI = f

p

do

frcm

2rp2 sin 0 dp

or

JBI = 2 r

f

a sin .0 d4)

J0

(13.563) p2 dp

or

(13.564) Ill =

23

f [f(4)]3 sin -, do.