r453sinthetaSolution 1 The endpoints of the region are 1 0 a

r=4/(5-3sintheta)

Solution

1) The endpoints of the region are (1, 0) and (1, p). dA = (1/2) r^2 d? = (1/2) (1 - sin ?)^2 d? A = (1/2) ? (1 - sin ?)^2 d? ˜ 0.356 2) r = 4 - 6 sin ?= 0 4 = 6 sin ? 2/3 = sin ? ? = 0.730 and 2.412 are the endpoints of the loop A = (1/2) ? (4 - 6 sin ?)^2 d? ˜ 1.764 3) We can use symmetry to find the area. The endpoints of the common interior are ? = p/4 and ? = 5p/4. If we find the area between those two values, the total area is twice that. (1/2) A = (1/2) ? (5 - 3 sin ?)^2 d? ˜ 25.125 A ˜ 50.251 4) Intersection points are ? = p/6 and ? = 5p/6. A = (1/2) ? [(2 - sin ?)^2 - (3 sin ?)^2] d? ˜ 5.196