37. Verify Green's theorem in the plane for f (3x2- 8y2) dx + (4y - 6xy) dy, where C is the boundary of the C
region defined by: (a) y = fx , y = x2 ; (b) x = 0, y = 0, x +y = 1 .
Ans. (a) common value = 3/2 (b) common value = 5/3
38. Evaluate f (3x +4y)dx + (2x --3y)dy where C, a circle of radius two with center at the origin of the xy
C plane, is traversed in the positive sense. Ans. - 87T
39. Work the previous problem for the line integral f (x2+y2)dx + 3xy2 dy. Ans. 127T
C 40. Evaluate f (x2-2xy)dx +(x2y+3)dy around the boundary of the region defined by y2 = 8x and x = 2
(a) directly, (b) by using Green's theorem. Ans. 128/5
(TT.2) 41. Evaluate f (6xy - y2) dx + (3x2 --- 2xy) dy along the cycloid x = 6 - sin 6, y = 1 - cos 6.