Will the area under a voltage vs. time graph of a battery discharge curve (with a constant current load) give the amp-hour capacity rating of the battery?
No, in your case it is not the area that tells you the capacity, but the length of time before the voltage is so low that the battery is considered discharged. For example, if the battery started out a 4.0 V, then decreased over 1.9 hours to 2.7 V when you considered it empty, the capacity is 1.9 hours times the discharge current. If the discharge current was 1.3 A, for example, then the capacity was (1.9 h)(1.3 A) = 2.5 Ah.
The area under the graph is proportional to the total energy the battery delivered. Actually its merely the time integral of volts. But you said that the current was constant, so volts is proportional to power, and the time integral of power is energy.
It's not the area under the curve but the partially discharged (final) level of the voltage that is important thing. The area gives you volt-hours and not ampere hours.
A manufacturer uses a constant current load and whilever the battery is able to produce above a "certain" minimum terminal voltage the hours that count the amps will keep ticking.
This, however does not mean that it will suit a specific target design. Here is an example of a discharge curve for a lightweight PP3 9V battery: -
Note that they define the end voltage and this means "end of test" because to drag more enrgy from the battery is going to either make it unsafe, unusable in 90% of target applications and render it impossible to recharge (if it were a rechargeable type). You have to set an end limit that is meaningful or you'd be conning the public.
Look at this AA battery: -
After just over 2 (and a bit) hours of 100mA discharge it's not a 1.5 volt battery any more. Another thing to watch out for is how they have used a constant current source - calculating ampere hour rating at 1mA gives you a better figure than at 100mA. Buyer beware and read the spec.
Take a look at the Duracell MN1500 battery and if you have any questions on this raise them. It should give you a good idea what a good supplier does to test their batteries.
Will the area under a voltage vs. time graph of a battery discharge curve (with a constant current load) give the amp-hour capacity rating of the battery
[emphasis mine, N.A.]?
No. Not even close. Even units of measurement would not match.
Here's how to get Ah rating for the battery from voltage vs. time chart. One needs one more piece of information: discharge current. It's usually written on the chart.
Let's say that the cutoff voltage for the battery in this chart is 6V. Under this voltage, we consider that the battery is depleted. At 8A discharge current, it will take roughly 20 minutes to discharge the battery down to 6V level. 8A * 0.3h = 2.4Ah (notice that the supplier advertised 3Ah).
(By the way, here's a somewhat related thread with a different set of charts.)