= Scalar surface integral over a cylinder with one tilted cap
{title2=$\int_S(\alpha y+z)\,dS=\pi R(H^2-\alpha^2R^2/2)+H\pi R^2\sqrt{1+\alpha^2}$}
Let $S$ be the full boundary of $x^2+y^2\leq R^2$, $0\leq z\leq H-\alpha y$, with $H>|\alpha|R$. The bottom contribution is zero by symmetry. On the top, the integrand is the constant $H$ and the <surface area element> is $\sqrt{1+\alpha^2}\,dx\,dy$. On the side, set $y=R\sin\phi$ and integrate in $z$ first: the result is $R[H^2-\alpha^2R^2\sin^2\phi]/2$. Integrating from zero to $2\pi$ gives the displayed <scalar surface integral>. No orientation signs enter; a scalar area integral is different from a <flux>.
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