I am interested in a specific question involving the computation of the volume (perhaps area is more appropriate?) of a semi-algebraic region defined by a ternary quadratic form and a ternary cubic form. The question can be phrased more generally as follows. Let
$$\displaystyle Q(x,y,z) = a_1 x^2 - a_2 y^2 + a_3 z^2, a_i > 0 \text{ for } i = 1, 2, 3$$
be an indefinite ternary quadratic form with negative determinant and
$$\displaystyle T(x,y,z) = \sum_{\substack{i_1, i_2, i_3 \geq 0 \\ i_1 + i_2 + i_3 = 3}} b_{i_1, i_2, i_3} x^{i_1} y^{i_2} z^{i_3}$$
be a ternary cubic form with real coefficients. Let
$$\displaystyle \mathcal{R}_1 = \{(x,y,z) \in \mathbb{R}^3 : Q(x,y,z) = 1, |T(x,y,z)| < 2\},$$
$$\displaystyle \mathcal{R}_2 = \{(x,y,z) \in \mathbb{R}^3: Q(x,y,z) = -1, |T(x,y,z)| < 2\},$$
and
$$\displaystyle \mathcal{R}_3 = \{(x,y,z) \in \mathbb{R}^3 : T(x,y,z) = \pm 2, |Q(x,y,z)| < 1\}.$$
Are there known techniques to compute the volume (preferably give explicit formulas) of these regions?