What does an asterisk (*) mean in the definition of an integral?

[find_the_fun]

In the definition of an integral what does the astrix (*) mean above the \(\displaystyle x_i\)? I got confused in class today because the prof used an astrix but just to mean the equation we had been talking about.

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Also, I sometimes see the top of the sigma being \(\displaystyle n-1\) instead of \(\displaystyle n\). I guess it doesn't really make a difference since \(\displaystyle n\) goes to infinity but why the difference?

 

[Nono713]

Re: What does an astrix (*) mean in the definition of an integral?

In this context (Riemann sums, I presume) it means "any $x_i$ within the subdivision interval" (this interval being a function of $\Delta x = \frac{b - a}{n}$, which tends to zero, and represents the width of the little rectangles you are using to approximate the integral). This is because you'll see that no matter what $x_i$ you pick within this interval (left point, right point, midpoint, some random point, ...) the Riemann sum will converge to the integral.

See here for a better explanation.

As for the $n$ versus $n - 1$ problem, have you checked that the bottom doesn't become $0$ when $n - 1$ is used? Those are all conventions and are equivalent (or at least, they should be).