MTH 212: Polar Curves (Sections 10.3, 10.4)

Example 1: fix r = 2, vary θ from 0 to 2π (circle)

PolarPlot[r=2,{theta,0,2Pi},PlotRange->{{-3,3},{-3,3}}]

Example 2: region 1≤r≤2, 0≤θ≤π/2

Show[
PolarPlot[{r=1,r=2},{theta,0,Pi/2},PlotRange->{{-3,3},{-3,3}}],
RegionPlot[1<r<2 ,{r,0,2},{t,0,Pi/2},PlotPoints->30]/.GraphicsComplex[a_,b__]:>GraphicsComplex[#1 {Cos[#2],Sin[#2]}&@@@a,b]
]

Example 3: cardioid r = 1-cos θ (using slider)

Manipulate[
PolarPlot[r=1-Cos[t],{t,0,tf},PlotRange->{{-2,2},{-2,2}}],
{tf,0.01,2Pi}]

Finding the arc length of the cardioid above

Integrate[Sqrt[(1-Cos[t])^2+(Sin[t])^2], {t,0,2Pi}]

Finding the area enclosed by the cardioid

0.5*Integrate[(1-Cos[t])^2, {t,0,2Pi}]

Example 4: Limacon r = 3+8 sinθ (“snail”)

Manipulate[
PolarPlot[r=3+8Sin[t],{t,0,tf},PlotRange->{{-12,12},{-12,12}}],
{tf,0.01,2Pi}]

Example 5: Example of a neat polar rose r = 1-2sin(3θ)

Manipulate[
PolarPlot[
1-2Sin[3t],
{t,0,tf},PlotRange->{{-4,4},{-2,4}}
], 
    {tf,.01, 2Pi}
]

Example 6: Rose r = cos(2θ) using PolarPlot and then ParametricPlot

PolarPlot[r = Cos[2t], {t,0,2Pi}, AxesLabel->{"x","y"}]
ParametricPlot[{x=Cos[2*t]*Cos[t],y = Cos[2*t]*Sin[t]},{t,0,2Pi}]

Finding the area of one leaf on the rose

2*Integrate[0.5(Cos[2t])^2, {t,0,Pi/4}]

Example 7: Plotting the cardioid and unit circle using parametric equations and a slider

Manipulate[
ParametricPlot[{{(1-Cos[t])Cos[t], (1-Cos[t])Sin[t]},{Cos[t], Sin[t]}},{t,0,tf}, PlotRange->{{-2,2},{-2,2}}],
 {tf,.01, 2Pi}
]

Finding the area inside the circle but outside of the cardioid

Integrate[0.5(1-(1-Cos[t])^2), {t, -Pi/2, Pi/2}]