Compute spatial integral using analog computing
In your particular case this makes little sense for an analog computer, as analog computers are at their best when dealing with time-dependent functions. It only makes sense as an educational problem. In analog computers time represents itself, everything else must be explicitly "coded."

You have basically two options:


Convert your space-based problem into a time-based problem, and start/stop the integration at the proper time points.


Note that any other function that depends on time would have problems with this transformation, so it might not be applicable to your example.


Discretize your space-based problem into an array of equations that are still applicable in the time-domain.


The second approach is a common one, and is just the application of Newton-Rampson style methodologies to the space domain.

The problem with this second approach is that it can lead to unstable analog implementations, as long-chains of feedbacks are sure to introduce dependencies on second-order analog effects (e.g., higher-frequency poles).

Analog computers are at their best when you let physics represent themselves, if you have to rely on discretization or other types of discontinuous approximations you are likely to run into problems.