3. Sketch r(0) = 1+2 cos(20) on your calculator for in the interval [0,27]. There should betwo large loops and two small loops.(a) What are the maximum and minimum values of the radius?(b) Find at least three O-values where r(0) = 0. Use exact positive values and radians.I(c) Write an integral that gives the area inside one small loop.(d) Write an integral that gives the area inside one large loop.

1st we will draw the graph of given function , then will find maximum and minimum value of r and values of θ If f is continuous and non negative on the interval α≤θ≤β with 0<β−α≤2π. Then area of the region bounded by the graph of r=f(θ) between the radial lines θ=α and θ=β is A=12∫αβr2dθhere equation of given curve r=1+2cos2θ a) we can use range of cos -1≤ cos2θ≤1-2≤2 cos2θ≤21-2≤1+2 cos2θ≤1+2-1≤r≤3 maximum value of radius = 3 minimum values of radius = 0 ( As r can not be negative ) when rθ=01+2cos2θ=0cos2θ=-122θ=2π3.4π3,2π+2π3θ=π3.2π3,4π3Area under curve A=12∫r2dθ =12∫1+2cos2θ2dθ =12∫1+4cos2θ+4cos22θdθ =12∫1+4cos2θ+21+cos4θdθ =12∫3+4cos2θ+2cos4θdθ =123θ+4sin2θ2+2sin4θ4 =3θ2+sin2θ+sin4θ4c) For small loop π3≤θ≤2π3 Area A=12∫π32π3r2dθ =3θ2+sin2θ+sin4θ4π32π3 =32×2π3+sin2×2π3+sin4×2π34-32×π3+sin2×π3+sin4×π34 =π2-334d)For larger loop -π3≤θ≤π3 A=12∫-π3π3r2dθ =3θ2+sin2θ+sin4θ4-π3π3 =32×π3+sin2×π3+sin4×π34-32×-π3+sin2×-π3+sin4×-π34 =32×2π3+sin2×2π3+sin4×2π34-32×π3+sin2×π3+sin4×π34 =π2+334