The integral is the area under the curve. Is there a similar notion for stochastic integrals?

For the geometric interpretation there is a nice result in the book by Karatzas and Shreve (chapter 3 2.29): Having a partition $\Pi$ with $0=t_0<t_1<\dots<t_n=T$ and defining $$ S_\varepsilon(\Pi) = \sum_{i=0}^{n-1} \big[(1-\varepsilon)W(t_i)+\varepsilon W(t_{i+1})\big]\big(W(t_{i+1})-W(t_i)\big) $$ then $$ \lim_{\|\Pi\|\to 0} S_\varepsilon(\Pi) = \frac 12 W^2(t)+(\varepsilon-1/2)t. $$ For $\varepsilon=0$ the right-handed side is the Itô integral $\int_0^T W(s)dW(s)$ and for $\varepsilon=1/2$ this is the Fisk-Stratonovich integral $\int_0^T W(s)\circ dW(s)$. So this is a nice interpretation for the stochastic integral as a geometric area. The only drawback is that the limit is in $L^2$-sense and we are dealing with stochastic processes (not just real functions) depending on a stochastic state $\omega$.

The general stochastic integral is first defined for simple (step-)functions exactly using the notion of area as in the Riemann case. For arbitrary functions approximations and limits are used. Thus again the notion of area is in the integral however somehow hided by the $L^2$-convergence and possible exclusions of null-sets where the integral can be defined completely arbitrary.

In classical analysis area and volume calculations are an important topic, but as Karatzas and Shreve point out in the introduction to their chapter 3, stochastic calculus was invented (and streamlined) for handling stochastic differential equations. So putting to much weight on the geometric interpretation might be misleading.