Understanding Physics Impulse: Integrated vs Average

[dgl7]

I have a question about physics in real life. Other than the fact that the area under a force versus time curve is a more accurate way to measure impulse because it's not an average like calculating impulse from a change in velocity is, is there any other reason that the integrated impulse would differ from the average impulse? Please let me know if I have phrased this oddly and you need clarification to answer my question. Thank you!

 

[SystemTheory]

As a thought experiment, you may go to this page:

http://www.nar.org/SandT/NARenglist.shtml

Open the pdf file for the Estes A10 engine and note the impulse is the area under the curve on the second page. Draw this impulse as an average (rectangle) and the big spike goes away, which is necessary to accelerate a model rocket up to stable velocity while on the launch rail. Both curves have the same total value if you take an impulse integral! The integral of the curve with the spike will accumulates greater impulse (area) in less time initially, causing a greater change in momentum in that region.