1550078481-Ordinary_Differential_Equations__Roberts_

124 Ordinary Differential Equations

So, the desired volume is Vy~ 27r(.4615936) = 2.900278.

Graphs of polar equations of the form r = a ± b cos e and r = a ± b sine

are called limai;ons. If lal < lbl, the limac;on is a closed curve with two loops.
(See Figure 3.1.) If ial = lbl, the limac;on is a single heart-shaped loop and
is , therefore, called a cardioid. (See Figure 3.2.) If lal > lbl, the limac;on is a
single, flattened, convex loop. (See Figure 3.3.)

Graphs of polar equations of the form r = a cos nB and r = a sin nB ( n 2: 2)

are called rose curves. If n is odd, the rose curve has n loops (petals); whereas,

if n is even the rose curve has 2n loops (petals). (See Figures 3.4 and 3.5.)

Graphs of r = a cos e and r = a sine are circles. And lemniscates are graphs of

polar equations of the form r^2 = a^2 cos 28 and r^2 = a^2 sin 2B. (See Figure 3.6.)

y 2 3 y

60°

x x

-3 -2 -1^2 3 4

300°

-2 -3

Figure 3.1 Limac;on r = l-2cosB Figure 3.2 Cardioid r = 2+2cosB

2 y 4 y

-4 2 4

0 x

• 1


• 2
  x
  -4
  -3
  -4

-5

-6 -4

Figure 3.3 Limac;on r = 3- 2sinB Figure 3.4 Rose Curve r = 4cos3B