Volume of intersections of spherical caps

Question

Suppose we have a unit sphere $\mathcal{S}^{d-1} \subset \mathbb{R}^k$, and for $\mathbf{u} \in \mathcal{S}^{d-1}$ and $\alpha \in (0, 1)$, define $$\mathcal{C}_{\mathbf{u},\alpha} = \{\mathbf{x} \in \mathcal{S}^{d-1}: \langle \mathbf{x}, \mathbf{u} \rangle \geq \alpha\} \subset \mathcal{S}^{d-1}.$$ I wish to compute the following quantities exactly, for small $k = 3, 4, 5, \dots$, where the input parameters are $\alpha$ and $\langle \mathbf{u}, \mathbf{v} \rangle$: \begin{align} \frac{\textrm{vol}(\mathcal{C}_{\mathbf{u},\alpha})}{\textrm{vol}(\mathcal{S}^{d-1})} \qquad \text{and} \qquad \frac{\textrm{vol}(\mathcal{C}_{\mathbf{u},\alpha} \cap \mathcal{C}_{\mathbf{v},\alpha})}{\textrm{vol}(\mathcal{S}^{d-1})} \, . \end{align} Are there any nice, exact formulas to work with here? I suppose ''unsolvable'' integrals without closed-form expression are inevitable, so perhaps some easy-to-evaluate formula to feed to mathematics software that allows me to get good, accurate numerical estimates of these quantities quickly?