Let C be the closed, piecewise smooth curve formed by traveling in a straight line between the points (0, 0, 0), (2, 1, 5), (1, 1, 3), and back to the origin in that order. Use Stokes' theorem to evaluate the integral. (11xyz) dx + (6xy) dy + xdz =Now we calculate the line integral using stokes' theorem :" J) ( 1xyz) dx + ( buy) ay + x dz = SS, curlF. ds = SS Thus , the normal vector is : n = = 0 7 - 2x - y +2=0 From the equation of the plane, - 2x- y+ 2= 0 the projection of surface on the xy-plane is the line 2xity = 0 by putting z= 0 in - 2x - y + 2 = 0 As on the xy plane, z= 0 , so the points (2, 1 , 5 ) and (1, 1, 3) becomes ( 2 , 1 ) and ( 1, 1 ) . Therefore, the domain is D = ( x , y ) : o sy s 1 , y s x = zy) since , equation of the line passes through ( 2, 1 ) and ( 0 , ) is : y - 1 _ 0-1 - y- 1 X - 2 0 - 2 21 - 2 = ! = 3 7 24 - 2 = 21-2 = 21= 24 Equation of the line passes through ( 1, 1 ) and (o, 0) is: = 0-1 3 4 - 1 = x- 1 7 2=y