restart;
# Computing the area enclosed by a parametric curve
f:=piecewise(0<=t and t<1,1,
             1<=t and t<2,t,
             2<=t and t<3,2,
             3<=t and t<=4,5-t);

g:=piecewise(0<=t and t<1,-t,
             1<=t and t<2,-1/t,
             2<=t and t<3,-(3-t)/2,
             3<=t and t<=4,0);

         

plot([f,g,t=0..4]);

A1:=-int(-1/x,x=1..2);

df:=diff(f,t);

A2:=-int(df*g,t=0..4);

f:=cos(t)*(2+sin(5*t));
g:=sin(t)*(2+sin(5*t));


plot([f,g,t=0..2*Pi]);

dg:=diff(g,t);

A1:=Int(dg*f,t=0..2*Pi);

value(A1);

evalf(A1);

df:=diff(f,t);

A2:=-int(df*g,t=0..2*Pi);