
THIRD STEP. Applying the Fundamental Theorem,

limit n=$$ $$ n i=1 1/2 \rho^2_{i} \Delta \theta_{i} = $$ \alpha \beta 1/2 \rho^2 d\theta.

Hence the area swept over by the radius vector of the curve in moving
from the position OP_{1} to the position OD is given by the formula

(A) area = 1/2 $$ \alpha \beta \rho^2 d\theta,

the value of \rho in terms of \theta being substituted from the equation of
the curve.

ILLUSTRATIVE EXAMPLE 1. Find the entire area of the lemniscate \rho^2 = a^2 cos 2\theta.

Solution. Since the figure is symmetrical with respect to both OX and OY, the
whole area = 4 times the area of OAB.

Since \rho = 0 when \theta = \pi/4, we see that if \theta varies
from 0 to \pi/4, the radius vector OP sweeps over the
area OAB. Hence, substituting in (A),

entire area = 4 $$ area OAB = 4 . 1/2 $$ \alpha \beta \rho^2 d \theta

= 2 a^2 $$ 0 \pi/4 cos 2 \theta d \theta = a^2;

that is, the area of both loops equals the area of a square constructed on OA as
one side.

EXAMPLES

1. Find the area swept over in one revolution by the radius vector of the spiral
of Archimedes, \rho = a \theta, starting with \theta = 0. How much additional area is swept over
in the second revolution? Ans. 4\pi^3 a^2 / 3; 8\pi^3 a^2.

2. Find the area of one loop of the curve \rho = a cos 2 \theta.  Ans. \pi a^2 / 8.

3. Show that the entire area of the curve \rho = a sin 2 \theta equals one half the area of
the circumscribed circle.

4. Find the entire area of the cardioid \rho = a(1 - cos \theta).
Ans.  3\pi a^2 / 2; that is, six times the area of the generating circle.

5. Find the area of the circle \rho = a cos \theta. Ans. \pi a^2 / 4.

6. Prove that the area of the three loops of \rho = a sin 3 \theta equals one fourth of the
area of the circumscribed circle.

7. Prove that the area generated by the radius vector of the spiral \rho = e^{\theta} equals
one fourth of the area of the square described on the radius vector.

8. Find the area of that part of the parabola \rho = a sec^2 \theta/2 which is intercepted
between the curve and the latus rectum. Ans. 8a^2 / 3.

9. Show that the area bounded by any two radii vectors of the hyperbolic spiral
\rho \theta = a is proportional to the difference between the lengths of these radii. 
