CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Example 5.26 2x Evaluate the line integral round the region bounded by the curve xydx  ydy y=x2 and x=y2 by use of Green’s theorem. Points of intersection are O(0,0) and A(1,1). 2 R Y 2 1 x O X 2 R 31 1 60 M=y N=-x In the special case when and   1 and Green’s theorem then states Mdx   Ndy R i.e. dxdy R   ydx xdy xdy ydx Therefore the area of the closed region A  dxdy 1  2 R C QUESTIONS 3x 2x C, 2 Q1. Evaluate where a circle of radius with center at  4 ydx  3 ydy the origin of the xy- plane, is traversed in the positive sense. Ans. 8 x x Q2. Evaluate around the boundary of the region defined 2xydx  y  3dy 2 2 by y2=8x and x=2. a) directly b) by using Green’s theorem
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Example 5.26 Evaluate the line integral xydx  2x ydy round the region bounded by the curve y=x2 and x=y2 by use of Green’s theorem. Points of intersection are O(0,0) and A(1,1). R Y 2 R 1 x 2 2 O X 1 31 60 In the special case when M=y and N=-x  1 and  Green’s theorem then states  Mdx  Ndy R i.e. dxdy R  ydx xdy  xdy ydx Therefore the area of the closed region A  dxdy  R 1 2 C QUESTIONS Q1. Evaluate 3x  4 ydx  2x 3 ydy where C, a circle of radius 2 with center at the origin of the xy- plane, is traversed in the positive sense. Ans. 8 Q2. Evaluate x 2 2xydx  x 2 y  3dy around the boundary of the region defined by y2=8x and x=2. a) directly b) by using Green’s theorem