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Given a two-dimensional area $A\subset\mathbb R^2$ lying in the $xy$-plane, i.e. $A:\mathbb R^2\supset U\to A\subset \mathbb R^2, (u,v)\mapsto (x(u,v),y(u,v))$, a straight extrusion along the $z$-axis defines a volume $V = A\times[z_\min,z_\max]$ and integration operator over this volume is simply $\int_V dV = \int\limits_{z_\min}^{z_\max}dz\int_Adx\,dy$ while the surface integral operator is $\int_{\partial V}d\vec n = \vec e_z\left.\int_A dx\,dy\right|_{z=z_\min}^{z_\max} + \int\limits_{z_\min}^{z_\max}dz\oint_{\partial A}d\vec n_A \quad(d\vec n_A = -\vec e_z\times d\vec r) .$ (Using the Kelvin-Stokes theorem $\oint_{\partial A} d\vec r = \iint_A (d\vec\sigma\times\vec\nabla)$ this can be used to prove the divergence theorem for extruded volumes by the way)

But let's now generalize this process to extrusion along a curved path $\vec\gamma:[0,T]\to \Gamma, t\mapsto \vec\gamma(t)$ defining $A(t):(u,v,t)\mapsto \vec\gamma(t) + \frac{d}{dt}\vec\gamma(t)\times(y(u,v),-x(u,v),0)$ (assuming $|\frac d{dt}\vec\gamma(t)|\equiv 1$) as a translation of $A$ both along and perpendicular to $\vec\gamma(t)$ to obtain the Volume $V_\gamma = \cup_{t=0}^{T} A(t).$

As a simple example, imagine a circle of radius $r$ extruded along a bigger (and perpendicular) circle of radius $R>2r$ which would yield a torus, while the extrusion along the $z$-axis would simply yield a cylinder.

What are the volume and surface integral operators now? I imagine it's something like $\int_V dV = \int_0^T dt D(t)\int_{A(t)}du\,dv$

And even more generally, what are the expressions if the area is tilted or rotated (torqued) along the way?

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    @nikmil Precisely, I'll update this into the question2012-11-15

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The simplest answer is simply to use the substitution rule: give a parametrization of your volume e.g. $F(x) = (A(x,y),z)$, then compute the Jacobi determinant (in our example $J(F) = J(A)$ and then $\int_V dV = \int_{x,y,z} |J(F)(x,y,z)| dx dy dz$. The Jacobi determinant can be easily computed with a CAS. This is always my way to derive Volume integral identities,