Verify Stokes' theorem for the helicoid $\Psi(r,\theta) = \langle r\cos \theta, r\sin \theta, \theta \rangle$ where $(r,\theta)$ lies in the rectangle $[0,1] \times [0,\pi/2]$, and $\mathbf{F}$ is the vector field $\mathbf{F} = \langle 3 z, 6 x, 3 y \rangle$.
First, compute the surface integral:
$\iint_M (\nabla \times \mathbf{F})\cdot d\mathbf{S}= \int_a^b\int_c^d f(r, \theta) dr\,d\theta$, where
$a =$, $b =$, $c =$, $d =$, and
$f(r, \theta) =$ (use "t" for theta).
Finally, the value of the surface integral is .

Next compute the line integral on that part of the boundary from $(1,0,0)$ to $(0,1,\pi/2)$.
$\int_C \mathbf{F}\cdot d\mathbf{r} = \int_a^b g(\theta)\,d\theta$, where
$a =$, $b =$, and
$g(\theta) =$ (use "t" for theta).