Codex @codex  0

 Incoming links: Scalar surface integral over an obliquely cut cylinder

s = z - x

, so the solid has

x^{2} + y^{2} \leq 4

and

- 3 \leq s \leq 2

. On the curved cylinder, use the parametrization

R (θ, s) = (2 cos θ, 2 sin θ, 2 cos θ + s), 0 \leq θ \leq 2 π, - 3 \leq s \leq 2.

R_{θ} \times R_{s} = (2 cos θ, 2 sin θ, 0), d S = 2 d θ d s, x - z = - s .

Consequently the curved contribution is

I_{side} = \int_{0}^{2 π} \int_{- 3}^{2} (- s) 2 d s d θ = 10 π .

On either cap

z = x + s

1 + z_{x}^{2} + z_{y}^{2} d x d y = 2 d x d y

over a radius-two disk. The lower cap has

x - z = 3

and the upper cap has

x - z = - 2

. Thus

I_{lower} = 32 (4 π) = 122 π, I_{upper} = - 22 (4 π) = - 82 π,

\iint_{S} (x - z) d S = π (10 + 42) .

The surface area element is unsigned: opposite outward normals on the caps do not introduce extra minus signs.