I'm currently studying numerical integration, and started with Monte Carlo Integration. Monte Carlo Integration states that
$$ I = \int_{a}^{b} h(y) dy = \frac{V}{N} \sum_{i=1}^N h(y_i), $$
where $y_i$ are samples drawn from a uniform distribution $\mathcal{U}(a, b)$ and the volume $V = b - a$ in the one dimensional case. The geometric interpretation of the formula is quite intuitive for me. We first compute the average likelihood of the function over the support domain and multiply it with the volume of the support domain. We basically construct a rectangle that has the same area as the one under the function, if we just draw enough samples.
Then I looked at importance sampling, which can be derived as
$$ I = \int h(y) dy = \int g(y) \frac{h(y)}{g(y)} dy = \mathbb{E}_{g(y)} \left[ \frac{h(y)}{g(y)} \right] = \frac{1}{N} \sum_i^N \frac{h(y_i)}{g(y_i)}. $$
In this case we draw samples from a helper distribution $g(y)$ instead of the uniform distribution like in Monte Carlo Integration. We are also not limited to a support. I understand the derivation from a formal point of view (and also the one of Monte Carlo Integration), but somehow I feel like the volume $V$ of the support is missing in this formula. Why is it not occurring in the formula, is it "hidden" somewhere? And how I can I imagine what's going on geometrically?