Use Stokes' Theorem to evaluate the surface integral?

Question

Use Stokes' Theorem to evaluate the surface integral where F=<zx^2, z(e^xy^2)-x, x*ln (y^2)> and where S is the portion of z=1-x^2-y^2 above the xy-plane with n upward.

Answer: -pi

x^2+y^2=1 r=1

x=cos t

y=sin t

z=0

dx=-sin t

dy=cos t

dz=0

integral of zx^2 dx+z(e^xy^2)-x dy+x*ln(y^2) dz from 0 to 2pi

This doesn't match the answer in the book. Can someone please correct me and show the work with steps?

Answer 1

F=<zx^2, z(e^xy^2)-x, x*ln (y^2)>

From the paramaterization,  F(r(t)),  <  0,  -cos t,  (cos t) ln (sin t)^2>, (everything multiplied by z vanishes)

r'(t) = i  (- sin t) dt   +  j  (cos t) dt + k (0)

Int [ F(r(t))*r'(t)]  dt  = Int [ i (0)(- sin t)  +  j (-cos t)(cos t)  +  k (cos t) ln (sin t)^2 (0)] dt

Int  [- (cos t)^2] dt

-  [t/2 + (1/4)sin 2t]  from 0 to 2pi, the second term vanishes

- (2pi/2 - 0/2)  = -pi