# Left and Right Hand Riemann Sum

I am stuck at Example of this Calculus Wikibook. In this example we will calculate the area under the curve given by the graph of $f(x)=x$ for $x$ between $[0,1]$. First we divide the interval into $n$ sub-intervals of equal width. So each sub-interval has width $\Delta x = \frac{1}{n}$. Wikipedia says Right Hand Riemann sum will be equal to:

$$x_i^* = 0 + i\,\Delta x = \frac in$$

I can't comprehend how zero is the first element in calculation using Right Hand Riemann Sum (RHRS). e.g. take this integral $$\int_0^1 x\,dx$$

Let us say I divide this into 4 equal sub-intervals, hence $\Delta x = \frac14$ and 4 sub-intervals will be $$[0,\frac14], [\frac14,\frac24],[\frac24,\frac34], [\frac34,\frac44]$$

Hence Left-Hand Riemann Sum (LHRS) meaning taking left points and RHRS means taking right points. That means $$LHRS = \frac14 [0 + \frac14 + \frac24 + \frac34]$$ $$RHRS = \frac14 [\frac14 + \frac24 + \frac34 + \frac44]$$

That means zero is coming as first element in LHRS but not in RHRS, then how come Wikipedia shows zero in RHRS ?

You're correct in your intuition, you've just made a mistake in the indexing. If you look at the link, it says that $i=1,2,...,n$. So, $i$ starts from $1$, not $0$. That means that your first interval is $x_1^*=\Delta x$, not $x_0^*=0$.
• @Arnuld Wikipedia is NOT saying that zero is the first element in the RHSR. It's saying the first element in the RHSR is $\Delta x$. If you look at the formula they gave, $x_i^*=0+i\Delta x$, it NEVER gives back $x_i^*=0$. Sep 10, 2018 at 16:16
• @Arnuld You're perfectly correct that zero is not the first element in the RHSR, and nowhere in the wikipedia page does it claim zero is the first element in the RHSR. It consistently says the first element in the RHSR is $(\Delta x)^2$ Sep 10, 2018 at 16:18