= Scalar surface integral over an obliquely cut cylinder
{title2=$\int_S(\alpha x-z)\,dS$}
Let $S$ be the whole boundary of $x^2+y^2\le R^2$, $a\le z-\alpha x\le b$, where $R>0$ and $a<b$. On the curved <cylinder>, the <parametrization> $(R\cos\theta,R\sin\theta,\alpha R\cos\theta+s)$ has <surface area element> $R\,d\theta\,ds$ and $\alpha x-z=-s$. Each planar cap has <area> $\pi R^2\sqrt{1+\alpha^2}$. Summing the side and both caps gives the <scalar surface integral>
$$
\boxed{\int_S(\alpha x-z)\,dS=\pi R(a^2-b^2)-\pi R^2\sqrt{1+\alpha^2}(a+b).}
$$
The <surface area element> is unsigned; cap orientation does not change the signs of the scalar integrand. This separates the geometric area factor from the field values and avoids mistaking a <scalar surface integral> for a <flux>.
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