= Solution
The <polynomial> <vector field> is smooth throughout the solid, so the <divergence theorem> applies. Differentiation gives
$$
\nabla\cdot\mathbf F=(2y+6x^5)+(-2y+4y^3)+1=6x^5+4y^3+1.
$$
The volume is invariant under both $x\mapsto-x$ and $y\mapsto-y$. Hence the <integrals> of the two odd terms vanish, and the <flux> is simply the volume. Using the circular cross-sections,
$$
\iint_S\mathbf F\cdot d\mathbf S=\pi\int_0^1 dz+\pi\int_1^2(2-z)\,dz=\boxed{\frac{3\pi}{2}}.
$$
The seams between the surface pieces have zero surface measure and contribute no extra term.
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