Parametrization of a weird shape

[Grayham1990]

Hey,

I'm currently preparing for my finals and I've found a bunch of questions I'm having trouble with,

This one is just a using Stoke's Theorem to determine a surface integral and I get an answer of zero on the RHS,

When I parametrize each surface of S I get a non zero result on the LHS of the equation.

Here is the question:

They gave a big hint to what G is and that F is actually the curl of H from the previous question,

H = (3y, -xz, yz^2 )

So S is made up of two surfaces and has 2 boundaries,

One boundary a unit circle in the x-y plane, the other a circle of radius 1/2

So I turned the boundary integral into two, one for each circle

I made my parameters thi(t) = (rcost, rsint, z), where r is either 1 when z is 1 or 1/2 when z is 2 and 0<=t<=2Pi

then just did the usual line integral, int H(thi) dot thi(t)' dt and got -4Pi for the unit circle and -5Pi/4 for the other

So together -21Pi/4

Which seems wrong to me, so I tried to check by doing the LHS surface integral

The cylinder param. was thi(u,v)=(ucosv,usinv,u) 0<=v<=2Pi, 1<=u<=2

did the usual thing, dot product of F(thi) with the cross product between the partials of thi wrt u,v

then for that part I got 16Pi,

The second part was the weird thing,

This is where I'm not sure at all, I did thi(u,v) = (u,v,u^2 + v^2)
did the usual stuff,

then after the dot product I let u=rcosq v=rsinq (q was theta)
and integrated for 0<=r<=1 and 0<=q<=2Pi

and got 7Pi

so total 23Pi no where near what I got on the RHS,

Does anyone know what I've done wrong?

Thanks alot