Explorations

A fun utility from Desmos:

https://www.desmos.com/calculator/hmn49xqtk8

Added my own twist to it with the quadratic on negative x.
One needs to think things through, because the visuals - and
the rounded numbers - can give a wrong impression.

At power 6 ie n=6, there is still only one value for x=0

Such a utility is what it is...

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source: Math if Fun

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Integration and summation over a function's range
might yield similar numbers, but they are not the same.
Summation is that of specific values, while integration
measures an area.

Below, for the ln function. Integration is the upper
term from zero, minus the lower from zero.  Because
0 to 1 is in the negative in this case, beginning at 1 ups
the numbers.

The actual integral for ln(x) is x(ln(x)-1). A whole other story!!
e^x is its own integral but addition at integers and area measure
give different results there, as well.