I want to show the animation of the polar equation r=cos(2theta) be plotted from theta=0 to 2pi

I want to shade the area where the two polar curves overlap. The first curve is r=2 and the other curve is r=2(1-cos(theta)). How do I do this?

I want to find the surface area of this parametric curve revolving around the x-axis. I was able to plot the 2D rendering but I want to show the plot so that I can see the surface area. I suspect that would be the 3D rendering.

x=cos(t), y=2+sin(t), 0<=t<=2pi, x-axis

I am trying to use this command

plot({0, 2*x^2, 2*x^2 - 2*x^3 + 8/3*x^4 - 4*x^5}, x = -10 .. 10, color = ["DarkGreen", "CornflowerBlue", "Burgundy"])

However, only the last function is showing up. 0 and 2x^2 is not showing up. I can't change the axes sizes because then the last function disappears.

I want to shade the function y=x*ln(x) on the domain of (0,1].