Oh, another wonderful use of integration:

Current sense probes.

If you do the old "scope probe grounded to clip" trick and wave that over a circuit, you're bound to see impulses and ringing. The ground loop isn't very particular at that size, so what you can do is, wind a small coil (maybe 10-20 turns of fine wire) and fit it in the end of a 6mm brass tube, which has been slotted up one side. Add a 50 ohm series termination resistor, and run coax up to the scope (with source termination, it's not necessary to terminate the scope end). Now, the brass tube shields stray E and B fields from your probe, except the ones that are axial with the tip. Wrap it with tape, and you can wave it around your circuit and sniff the currents flowing through components, traces, even individual vias!

But you're not actually measuring current. You're measuring EMF. Which is the derivative of current. So, you can integrate it, and now instead of impulses, you have steps -- corresponding exactly to the current steps in your circuit! Of course, it's not calibrated, but knowing the general proportions and how the signal varies with distance, you can get a feel for how much current is flowing in a circuit, and where.

Tim