Let S be that portion of the plane −12x +4y +3z = 12 projecting vertically onto the plane region (x −1)^2 + y^2 ≤ 4. Evaluate...

Question

Evaluate

a) the area of S

b) ∫∫z dS

c) ∫∫(x 2 + y 2 + 3z)dS

Showing what I've tried to do:

1. dS = √(12² + 4² + 1)dxdy = √(161)dxdy. Not sure if this is right.
2. For b) and c) I substitute z = 4 + 4x - (4/3)y, right.
3. I parametrized the circle on the xy-plane with x(θ) = 1 + 2cos(θ), y(θ) = 2sin(θ). Idk if this is useful.
4. I used nΔS = (n|N| / N∙k)ΔA, trying to relate the surface to the circle. I got ΔS = (13/3)ΔA. Idk if this is legal math.
5. Sketch: https://i.stack.imgur.com/IgvTs.jpg -- I made a small sketch with the circle on the xy-plane, a diagonal plane with a transformed circle on it, a normal vector to that plane and the angle ϕ from the z-axis to the normal vector of the plane.

Source: 6B-10* https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-b-flux-and-the-divergence-theorem/problem-set-11/MIT18_02SC_SupProb6.pdf