4 Finding areas with Green's theoremA planimeter is a mochanical device for measuring the area ofan arbitrary two-dimensional shape. It corsistsof two linkages, with one end fixed, one rotational or linear joint, and the other end used to trace the shape.The tracing arm has a measuring whoel which rotates around the axis of the arm. One way of understandingits function is through Green's theorem.In this question we will learn how Groen's theorem can be used to calculate an area, and apply ittoanexample for which the usual methods don't work.The area of a region Ris calculated by the double integralArea(R)=K1dAwhile Green's theorem states thatcPdx+Qdy=x(delQdelz-delPdely)dAwhere the left-hand side represents the line integral ofF=P(x,y)i+Q(z,y)j around C, the boundary curveofR with an anticlockwise orientation.(a)We can calculate area with a double integral if the integrand equals one.Ifwe choose P=0 then Q-, will result indelQdx-apdelj-1.Ifwe choose Q=0 then P=will result inaqdx-agdely-1.Another choice that gives ggdel-deldel-1isP=and Q=(b) Groen's thorem can change a double integral for area into a line integral around the boundary curvefor an appropiate vector field. Choose oae of the vector fields from (a)to calculate the area of thefollowing shape:which has boundary curve parametrised byr(t)=(sin(t)+12sin(2t)+23sin(3t))i+(32-12cos(t)-cos(2t))j,0t<2you may use the following identities which hold ifm and n are integers mis not equal ton 02cos(mt)cos(nt)dt =0 02sin(mt)sin(nt)dt=0