**Problem : **
Plot the polar curve given by *r*(*θ*) = cos(2*θ*) for *θ* = 0 to 2*Π*.

Figure %: Polar Plot of *r*(*θ*) = cos(2*θ*) for *θ* = 0 to 2*Π*

**Problem : **
What is the area contained within the region bounded by *r*(*θ*) = cos(2*θ*) from
*θ* = 0 to 2*Π*? You may use that cos^{2}(*θ*) = (1 + cos(2*θ*))/2.

(cos(2θ))^{2}dθ | = | dθ | |

= | θ + | ||

= | , |

exactly half the area of the unit circle in which it is contained!

**Problem : **
Find the area bounded by the graph of the *cardioid* defined by
*r*(*θ*) = sin(*θ*/2) for *θ* = 0 to 2*Π*, using the
identity sin^{2}(*θ*) = (1 - cos(2*θ*))/2.

Figure %: Polar Plot of *r*(*θ*) = sin(*θ*/2) for *θ* = 0 to 2*Π*

sin^{2}dθ | = | dθ | |

= | θ - sin(θ)) | ||

= |

once again equal to half the area of the unit circle in which the region is contained!

Take a Study Break!