Utilize either polar or rectangular coordinates to delineate three symbolic representations, exhibiting creativity, employing polar or parametric coordinates. Employing Desmos, GeoGebra, graphing calculators, or manual illustration, demonstrate the constraints, ensuring to enumerate the equations utilized for plotting the curves. Subsequently, elucidate the characteristics of your graphs. $($ex$.$

Amanda desires to adom herself with a temporary tattoo featuring her favorite geometric shape. This shape corresponds to the region enclosed by the polar curve $r=2cos(3).$ To ensure the tattoo's accuracy, the artist requires knowledge of the ink quantity necessary for its creation. Determine an expression involving integrals that yields the total area of the shaded region illustrated below. Subsequently; compute the area.

Consider a particle whose trajectory in the $xy-$plane is given by the parametric curve defined by the equations $x(t)=t^4-4t^2,y(t)=t^2-2t,$ for $-3t3.$ Show all your work to receive full credit.

a$.$ Is there any value of $t$ at which the particle ever comes to a stop? Justify.

b$.$ For what values of $t$ does the path of the particle have a vertical tangent line?

c$.$ What is the lowest point $(x,y)$ on the curve?

d$.$ At what values of $t$ does the particle pass through the origin?

e$.$ The graph of the curve traced by these parametric equations is shown below. Find an expression for the length of the closed loop marked in the graph.