Problem 3

Points={{0,0},{0.18,0.1},{0.32,0.23},{0.47,0.46},{0.46,0.51},{0.5,0.5},{0.24,0.32},{0,0}};
area=0;
area2=0;
For[i=2,i≤ Dimensions[Points][[1]],i++,
area+=(Points[[i,1]]+Points[[i-1,1]])/2 (Points[[i,2]]-Points[[i-1,2]]);
area2+=Points[[i,1]] (Points[[i,2]]-Points[[i-1,2]]);
];

area

area2

x[t_]:=Sin[t] Cos[2t] Sqrt[3]
y[t_]:=Sin[3 t] Cos[2t] Pi/4
NIntegrate[-x[t]y'[t],{t,0,Pi/4}]