p. 692, #6, 22, 38

Clear[r]

r[t_]:=1+Cos[t]
PolarPlot[r[t],{t,0,2Pi}]
NIntegrate[1/2*r[t]^2,{t,0,Pi}]

#22

r[t_]:=2Cos[t]-Sec[t]
PolarPlot[r[t],{t,-1.2*Pi/4,1.2*Pi/4}]

Solve for r[t]=0 to find the intersection points of the loop:

Solve[r[t]==0,t]

PolarPlot[r[t],{t,-Pi/4,Pi/4}]
NIntegrate[1/2*r[t]^2,{t,-Pi/4,Pi/4}]

r[t_]:=1-Cos[t]
s[t_]:=1+Sin[t]
Solve[r[t]==s[t],t]

Solve only finds those intersections that occur at the same parameter value. Other intersections occur, however

Solve[r[t]==0,t]
Solve[0==s[t],t]

p1=PolarPlot[{r[t],s[t]},{t,0,2Pi}];
p2=ListPlot[{{r[7Pi/4]Cos[7Pi/4],s[7Pi/4]Sin[7Pi/4]},{r[3Pi/4]Cos[3Pi/4],s[3Pi/4]Sin[3Pi/4]},{0,0}},PlotMarkers->{Automatic,Large}];
Show[p1,p2]