Although we implicitly use the definition of the Riemann Sums every time we cacluate a definite integral is it actually possible to calculate the majority of integrals via Riemann Sums?

Example:

this is easily able to be calculated using the rieman formula (I omit the details)

But what about another fairly elementary integral? How would we go about using the above defintion to find say

?

Is it actually feasible using just summing techniques (besides obviously integrating) to calculate ?

Even if someone could do the above sum (which Im sure someone can) what about a harder one like . And what about all of this using the formal Riemann-Stieltjes definitions?

A similar argument brings us to ask if we can practically find say

using only defintions?

Note: I am not asking whether or not we should use these defintions to find integrals and limits. What I am asking is if someone said compute using only Riemann sums would it be feasible?

Any input/discussion would be appreciated :)