Calculating cost of servicing incremental number of users?

[MitchT]

Putting this is Science as there's no Maths forum. A science bod might be able to help...

Scenario:

I launch an app which costs 5p per month per user to operate.
On Day 1 it has no users.
At the end of Year 1 it has 1,200,000 users.

How much does it cost to run over Year 1, bearing in mind that the users increase gradually over the year?

Consideration:
It doesn't jump to 100k users at the end of Month 1, and 200k users at the end of Month 2, etc., so it's not a case of saying 5p x 100,000 to find out Month 1 costs as you don't have 100,000 users for all of Month 1... 200,000 users for all of Month 2... etc.

Thanks in advance!

[BadBob]

I'll have a stab:
Assuming users are added at a constant rate the graph of number of users vs. time is a straight line, the right-hand end at 1,200,000. Visualise the graph and note that half the area is above the line and half below. The area below is what you have to pay support on.
If you have 1,200,000 users all year it would cost you 1,200,000 * £0.05 * 12 = £720,000. But from the graph you know you only have half that cost, £360,000 by the end of year 1.
It gets trickier if you have a different growth model, such as accelerating through word-of-mouth, I think that's what calculus can do but I've forgotten how to do it.
Good luck getting those 1.2 million users in just a year :-)

[MitchT]

Thanks - makes perfect sense. Of course, it's more likely to start as a fairly flat curve and then get steeper as usage picks up. it's one of those things that'll either fall flat on its face or gain so many users so quickly that were crapping ourselves over the hosting bill!