David Fontaine <dav### [at] faricynet> wrote
> Okay, I sort of follow al this, but integral outputs a function, right?
What's
> the different x-values of the function represent?

'x's in the result of an integration usually mean you didn't
actually sum from a specific x value to another specific x, so
you get a formula instead of a definite value for the integral.
The formula then has variables in it which you can use
when you decide where among the x's the 'actual' summing
is to be done.
(Instead of a definite value which you would get if you started at a
definite place and ended at a definite place.)

>
And what would be the point of
> finding the average of all points in a planet's orbit? it'd just be the
center
> of the ellipse

oh no. the average was just one example of an 'integration'. the planet
was another.

Using the planet example, you might say: start the planet "here"
at coordinates x0, y0 going at velocity vx0, vy0. Say the sun is at 0,0.
Then, knowing the planet's position  you can calculate the force the
sun has on the planet. From that you figure (using Newton's laws for
example) how the planet's velocity will change. Add that change to
the planet's velocity. That new velocity takes it to a new position.
There at the new position, it'll feel a different gravity force, and change
its velocity again, and move with that new  velocity to a new position,
where it feels a different gravity force, and so on, over and over,
tirelessly adding the effects on the planet's position.
But between 'here' and 'there' the gravity is constantly  changing your
velocity so you can't just say it will go with a constant velocity from
'here'
to there'. You jump ahead in small steps so the velocity doesn't change
"much" so where you calculate you will end up at the end of a small
time is pretty close to where you will actually end up.
If you make your steps infinitesimally small you can apply theorems and
rules of 'integral calculus' to get the result. And if the equations for the
forces and motion are 'nice' enough, those rules can lead to a
simple formula which might depend on where you start (x0, y0) and
initial velocity (vx0, vy0).



if there are variables in what you are summing, then the integral could
be a function, but if it is summing numbers, no its not much of a function.

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