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 ?
Best Answer
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 $x_0^*=0$.