10.3 Areas and integrals involving polar coordinates 549in the negative direction. We never have negative areas, but we can
subtract areas because areas are numbers. In case the path of P(t) is
the closed curve C of Figure 10.36, I is the area of the region bounded by
C. If the path of P(t) is the curve obtained by reversing the direction
of the arrows in Figure 10.36, then I, is the negative of the area of the
region enclosed by the curve. Except in cases where the vector from
0 to P(t) always rotates in the same direction, the number 12 in (10.34)
is usually less interesting than I,. It is the sum of the areas of all of the
regions swept over by the rotating vector.
Problems 10.39
(^1) Find the area of the region bounded by the cardioid having the polar
equation
p = a(l + cos 0).
l1ns.: 3aa2/2.
(^2) Using the polar equation p = 2a cos 0 of a circle of radius a, and noting
that a particle with polar coordinates (p,4) traverses the circle once as q5 increases
from --7r/2 to it/2, work out the familiar formula for the area of the circular
disk bounded by the circle.
3 The graph of the polar equation p2 = a2 cos 20 is the lemniscate shown
in Figure 10.171. Find the area of the bipartite set which it bounds .4ns.: a2.
(^4) Find the area of the region bounded by the graph of p = cos o. 14ns.: 1.
5 Supposing that a > 1 and that n is a positive integer, find the area of the
region bounded by the polar graph of the equation
p = a+cosno.
.Ins.: 7ra2 + it/2.
6 Use integration to find the area 111 of the smaller region which the line with
rectangular equation y = x slices from the interior of the circle having the polar
equation p = 2a cos 0. Then calculate 11, from the fact that the interior of
the circle is the union of the interior of an inscribed square and four slices each
having area A,. Make the results agree.
7 Sketch a polar graph of the equations
p=4+cost, 10 =2sint
and make a rough estimate of the area of the region enclosed by the graph. Then
calculate the area.
8 Set up an integral for the area -4 of the region bounded by the ellipse
having the standard polar equation
ep
P=1 - e cos 0
and show that the result can be put in the form
- p2^1 (p
Jo (e''-cos4,)2d