### Graphs of Functions in Polar and Rectangular Coordinates with an Area Tracer

The function \(f\) is geometrically interpreted
as a curve in the plane in two ways:
first as its graph \(y=f(x)\) in rectangular (Cartesian) coordinates
as the locus of points \(\big(x, f(x)\big)\),
and second as its graph \(r=f(\theta)\) in polar coordinates
as the locus of (rectangular) points \(\big(r\cos(\theta), r\sin(\theta)\big)\).
A bubble traces out corresponding points on these two graphs
between \(\theta_{\text{min}}\) and \(\theta_{\text{max}}\),
shading in the area “under” the polar graph,
green for positive area and red for negative area.