Equivalence of the nonstandard analysis integral and the Riemann integral

Question

I have a question about the definition of the integral in nonstandard analysis. The definition that I've usually seen is this: given a function $f(x)$ that you want to integrate from $a$ to $b$, you define a function $$S(\Delta x) = \sum_{x = a}^b f(x) \Delta x,$$ which you then extend to a hyperreal function through the transfer principle. Then the integral is defined as the standard part of $S(dx)$, where $dx$ is infinitesimal. The definition feels fine to me except that it looks like it is equivalent to defining an integral through left sums, which is famously not the same as a Riemann integral. So my question is what's wrong here? Is this definition that I gave not the usual one used in nonstandard analysis? Is my argument that this is equivalent to left sums flawed? Is the nonstandard integral actually different than the Riemann integral?

Answer 1

As mentioned in a comment, this definition of the nonstandard integral is only used for continuous functions, in which case the integral based on left sums is the same as the Riemann integral.