Verify Stoke's theorem for this surface

[s3a]

Homework Statement

The problem and its solution are attached in TheProblemAndSolution.jpg.

Homework Equations

Stoke's theorem: ∮_C F ⋅ dr = ∮_C (F ⋅ T^{^}) dS = ∫∫_S (curl F) ⋅ n^{^} dS

The Attempt at a Solution

In the solution attached in the TheProblemAndSolution.jpg file, I don't understand what's going on with the integral that has |n^{^} ⋅ k^{^}| on a denominator.

Could someone please add the steps that are skipped by the solution?

 

[STEMucator]

An easier way would be to say:

$$\oint_C \vec F \cdot d \vec r = \iint_S \text{curl}(\vec F) \cdot d \vec S = \iint_D \left[(z^2 + x) \hat i - (z+3) \hat k \right] \cdot (\vec r_x \times \vec r_y) \space dA = \iint_D \left[(4 + x) \hat i - 5 \hat k \right] \cdot (\vec r_x \times \vec r_y) \space dA$$

Where $\vec r(x,y) = x \hat i + y \hat j + 2 \hat k$.

A simple switch to polar co-ordinates afterwards would clean that up nicely.