Jo EXAMPLE 4 Evaluate o 5y2 dx + 12xy dy, where C is the boundary of the semiannular region D in the upper between the circles x2 + y2 = 1 and x2 + y2 = 16. SOLUTION Notice that although D is not simple, the y-axis divides it into two simple regions (see the figure). In p coordinates, we can write D = { (r. 8 ) | 1 5 r s 4, 0 5 0 ST ). Therefore Green's Theorem gives $ 5y2 dx + 12xy dy =(12xy) - - ( 5x2 ) dA 2y dA ( r sin ( 8) )r dr de - 2 / " sin ( @) de 2 dr = 2 -cos ( 0 ) 104 3 X