= Solution
Introduce the second moment <tensor> $M_{ij}=\int_Vx_ix_j\,d^3x$. The <spherical shell> is preserved by every <rotation>, so this is an <isotropic tensor>, $M_{ij}=A\delta_{ij}$. Its <trace> determines the constant:
$$
3A=M_{ii}=\int_Vr^2\,d^3x
=4\pi\int_{R_1}^{R_2}r^4\,dr
=\frac{4\pi}{5}(R_2^5-R_1^5).
$$
Contracting with the components of the two unit <vectors> gives
$$
\boxed{\int_V(\mathbf x\cdot\mathbf n_1)(\mathbf x\cdot\mathbf n_2)\,d^3x
=\frac{4\pi}{15}(R_2^5-R_1^5)(\mathbf n_1\cdot\mathbf n_2)}.
$$
This is the second-order case of <radial tensor moments in three dimensions>.
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