Green's theorem

Hello!
5b. Green's theorem gives us a possibility to compute the area of a plane region integrating along its boundary. Actually, it can help for more complex tasks then computing area. There are two (and more) forms of that integral,
|oint_C x dy| and |oint_C y dx|, where C is the bounding curve.
In our case y better suits as the independent variable, so compute oint_C x dy. The curve consists of two parts, and the integral is the sum of two integrals, int_(C_1) and int_(C_2), where C_1 is the segment and C_2 is the semi-circumference. Note that to get round the boundary in the right direction, we have to integrate over C_1 from the larger y to the smaller.
The rest is simple,
int_(C_1) = int_1^(-1) x dy =int_1^(-1) 1 dy = -2,  and  int_(C_2) =int_(-1)^(1) x dy =int_(-1)^(1) y^2 dy = 2/3.
So the area is |-2+2/3| = 4/3.
 
The question 5a should be asked separately. If you would ask it again, please make clear what symbol is after (x+2y^2).  (-j) which means unit vector of the y-axis?