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Definition 2 is from the textbook of Stewart calculus

The area $A$ of the region $S$ that lies under the graph of a continuous function $f$ is the limit of the sum of the areas of approximating rectangles. $$A = \lim_{n\to\infty}R_n=\lim_{n\to\infty}[f(x_1)\Delta x+f(x_2)\Delta x+\dots+f(x_n)\Delta x]$$

When ${n\to\infty}$, we have $\Delta x\to0$, so $$A = \lim_{\Delta x\to0}R_n=\lim_{\Delta x\to0}[f(x_1)\Delta x+f(x_2)\Delta x+\dots+f(x_n)\Delta x]$$ when $\Delta x\to0$, we have $$\lim_{\Delta x\to0}f(x_1)\Delta x=\lim_{\Delta x\to0}f(x_2)\Delta x=\dots=\lim_{\Delta x\to0}f(x_n)\Delta x =0$$

Then we use Limit Rule, $$A =\lim_{\Delta x\to0}f(x_1)\Delta x+\lim_{\Delta x\to0}f(x_2)\Delta x+\dots+\lim_{\Delta x\to0}f(x_n)\Delta x=0 $$

I know the answer is wrong, but I don't know where the error is. Can someone help me out?

Tian Vlasic
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Andrew Li
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  • Note that at the same time $n\to +\infty$. You cannot just keep $n$ fixed and let $\Delta x\to 0$, because tha latter is not true if $n$ is fixed. – Gary Jun 26 '23 at 04:41
  • Does $$\lim_{\Delta x\to0}f(x_1)\Delta x $$exist? Or we don't know if it exists? – Andrew Li Jun 26 '23 at 04:56
  • https://math.stackexchange.com/questions/4725643/limit-laws-be-applied-to-infinitely-many-terms?noredirect=1#comment10013973_4725643 – Andrew Li Jun 26 '23 at 14:23

1 Answers1

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Always remember Limit is an operation, or a "short-hand" for a group of more simpler elementary operations (just like we use methods in programming languages to represent a group of simpler statements collectively together).

Here, Limit $\Delta x \to 0$ is a short-hand for "keep on decreasing $\Delta x$, but at the same time keep on increasing $n$, such that $(\Delta x)(n)$ remains constant .

You have $\Delta x$, you sample $2$ times more values for $f(.)$ at the same time.

Your mistake was assuming $n$ stays constant, no matter how much small you make $\Delta x$.

Hope this helped! :)

Gary
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Sagnik Chakraborty
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