normal distribution

A game session has a mean of 0, and standard deviation of 3.

I know the chance of falling between mean and +1 standard deviation is 34%
I know the chance of falling between mean and +2 standard deviation is 48%

How do I find out the average win size for game sessions between mean and +1 standard deviation? And between 0 to 2 standard deviation?

In case the above is not clear. Let say I will play this game session once.
If the result is between 0 and +1 standard deviation I take that winning home.
If the result is outside that range, my partner will pick up the tab.
What is my EV per session?

(honest play, not play until I am in that range and stop)

Just integrate over the normal distribution.

average win between +0 and +1 sd: 0.47.


average win between +0 and +2 sd: 1.03.

awesome, thanks. I was thinking there might be a look up chart, like the one for looking up cumulative normal distribution.
Actually I chopped up the area between +0 and +1 into 20 vertical bars and added up the area.... only off by 0.04 from your numbers

And here's a nice closed-form expression for your average EV over 0 to +N standard deviations (N*sigma):

EV = sigma*(1-exp(-N^2/2))/sqrt(2*pi)