Given any function in the slope-intercept form, be it linear or non-linear, the area between the curve and the x-axis between two points can be easily approximated with the Riemann Sum using a For loop.

This is a simple pseudo-code example, where Y1 is the function and Y1(I) is the value of Y1 at X = I:

N = 0
For (I, Left Limit, Right Limit, Step Interval)
    N = Y1(I) + N
End
Display N

To determine the step interval, use this:

Step Interval = (Right Limit - Left Limit) / Number of Steps

To improve accuracy, there are multiple ways of using this approximation, which distinguish between which end of each step to use.  To adjust the limits and intervals for the desired method, use this code:

If LRAM (using left end)
    Right Limit = Right Limit - Step Interval
End If
If MRAM (using middle of step, often most accurate)
    Left Limit = Left Limit + (Step Intverval / 2)
    Right Limit = Right Limit - (Step Intverval / 2)
End If
If RRAM (using right end)
    Left Limit = Left Limit + Step Interval
End If

For anyone wishing to have a complete program for a TI-83 or TI-84 calculator, use my Riemann Sum Program.

Please note that this is an approximation. You can find exact answers using integrals.

As the first letter in the acronym SMTP stands for simple, that’s what I was expecting when I decided to use the Google Apps SMTP server to send out notifications for Xusix.  In the end, after three days of repeatedly rewriting the mailer functions, I discovered the reason nothing was sending.  GoDaddy shared hosting doesn’t allow SMTP connections.  At all.  They simply block anything that appears to be sent through that protocol, or uses a common SMTP port, leaving me to the only remaining option: using GoDaddy’s own mailer.  The problem with this is that their spam filter is so aggressive, few messages ever make it through.  And so, notifications on Xusix, although programmatically work, will likely never reach an inbox.