Let $X$ denote a (compact) Calabi-Yau 3-fold, and suppose $D_I$ denotes a basis of divisors on $X$ (these are classes in $H_4(X, \mathbb{Z})$) and $\omega_I$ denotes a basis of $H^2(X, \mathbb{Z})$ (Poincaré dual of $D_I$). Further let $J$ denote the Kähler form of X.
I want to write a formula for the volume of a divisor $D_I$ as an integral over the entire Calabi-Yau. Of course, by definition one has
$$\text{vol}(D_I) = \frac{1}{2}\int_{D_{I}}J\wedge J$$
so one could extend this in principle to the entire $X$ by writing something like
$$\text{vol}(D_I) = \frac{1}{2}\int_{X} J \wedge J \wedge \delta_{D_I}$$
where $\delta_{D_I}$ is a real 2-form which localizes the integral over $X$ to an integral over the codimension-1 surface defining $D_I$.
Now, I can decompose the Kähler form in the basis $\omega_I$ as
$$J = \sum_{I=1}^{h^{1,1}}t^I \omega_I$$
and plug this into the expression for $\text{vol}(D_I)$ above to yield
$$\text{vol}(D_I) = \frac{1}{2}\sum_{J,K=1}^{h^{1,1}} t^J t^K \int_{X}\omega_J \wedge \omega_K \wedge \delta_{D_I} \qquad \text{($\#$)}$$
My hope is/was to be able to write the integral in terms of the triple intersection numbers of the Calabi-Yau. But those are naturally quantities of the form
$$\int_{X} \omega_I \wedge \omega_J \wedge \omega_K$$
How do I relate ($\#$) to the triple intersection number?