Analog Integrator Sampling Rate What would be the effective sampling rate of an op amp analog integrator (such as the one described here ) taking in a rapidly varying signal be? Ie what time resolution would such an integrator have? 

The need for this to solve a practical problem has passed, as a different method of testing was ultimately selected, and so I intend this question to be as theoretical as possible, but in order to understand situation better I shall provide the original motivation. I was attempting to use a load cell in order to measure the momentum change of an object colliding with my load cell by taking the force reading of output of the load cell over time and numerically integrating the result. Unfortunately, my data logger only sampled at 1Khz and the impulse event was estimated to take only a few ms, thus providing much too small time resolution to accurately integrate afterwards. So, My thought was to hook the load cell output directly to an analog integrator and I should get a step function output that would read off to me a value proportional to the momentum change. From the transfer function of 1/s, it would seem as though I could do it for any signal regardless of how rapidly changing the signal is, however there must be some practical limit on how fast the integrator can respond. Can I assume a transistor switching speed (on the order of many Mhz and thus much faster than needed here) for the time resolution?
Sampling rate is only defined within discrete-time systems. So, there's really no sampling-rate equivalent that your integrator has.

What your integrator does have is finite bandwidth. It, for example, will not be sensitive for voltage variations at 2.4GHz.

The bandwidth off that circuit involves the characteristics induced by modeling the circuit based on ideal components, but also on limitations such as unintended (parasitic) inductance and capacitance of components and leads, and, most importantly: by the bandwidth of the opamp itself.

You can buy opamps with only a few Hz of bandwidth, you can buy them with multiple GHz of bandwidth, and very many values in between. Each of these have different strengths and disadvantages, so for a measurement system, choose wisely, based on an understanding of the requirements of your measurement system and signal, and at least a solid understanding of electronics and linear electrical networks; I'm not convinced designing a measurement integrator is trivial.

Generally, you might want to take a step back and actually ask over at electronics.SE how to measure a specific signal, describing exactly and in detail what you are observing, rather than assuming an integrator is the solution here. It might be - but it's equally likely you're not on the right track.