2) Consider the area of the butterfly-shaped region enclosed by fr (x) = k sin(kx) and gk (x) = 2k - k sin(kx) on the interval [0,"]. i) Graph the regions for k = 1 and k = 2. Formulate the areas of the enclosed regions. ii) Describe the shapes of the areas as k - co. iii) What is the area Ak between fr (x) and gh (x)? Prove your response. iv) Let the perimeter of the region with area Ax be denoted by Pr. What is lim PK? K-+00