The function r = 1 - sin
θ
can be graphed in the r,
θ
plane to help graph the cardioid in the
x, y plane.
The graph of the function in the r,
θ
plane can be derived by evaluating 1 -
sin
θ
for various
θ
’s from 0 to 2
π
.
By seeing how 1-sin
θ
behaves in the r,
θ
plane, the function of 1-sin
θ
can be graphed in
the x, y plane.
To graph the cardioid, start from
θ
= 0 and plot increments of
π
/2 all the
way to 2
π
, which is derived from the graph above (
θ
= 0, r = 1;
θ
=
π
/2, r = 0;
θ
=
π
, r =
0;
θ
= 3
π
/2, r = 2;
θ
= 2
π
, r = 1).
Essentially, by making
θ
range from 0 to 2
π
, it is
possible to graph one period of r = 1 - sin
θ
, which can be derived from the graph in the x,
y plane.
In addition to graphing the points in increments of
π
/2, it must be observed that
the r (of r = 1 - sin
θ
) decreases from
θ
= 0 to
θ
=
π
/2, increases from
θ
=
π
/2 to
θ
=
π
,
increases from
θ
=
π
to
θ
= 3
π
/2, and then decreases from
θ
= 3
π
/2 to
θ
= 2
π
, to trace out
the graph of the cardioid. Besides graphing the cardioid, the function r = 1 should be
graphed; this function is just a circle with radius one, with the center at the origin, in the
x, y plane.
The first part of the problem requires finding what percent of the cardioids enclosed area
lies above the x-axis.
To do this the area of the cardioid from 0 to
π
(which lies above the
x axis) must be divided by the entire area of the cardioid from 0 to 2