How can I visualize [the integrals below](https://betterexplained.com/articles/definitions-of-e-colorized/)? Can someone draw on the chart and point to where $\int 1 \, dx = x$, $\int x \, dx = \frac 12 x^2 $, ... are ? I don't know [which of these charts](https://betterexplained.com/articles/a-visual-guide-to-simple-compound-and-continuous-interest-rates/) are more intuitive, and I'll copy and paste two. Is this called a Bar Chart?
[![enter image description here][1]][1]
> From a calculus perpsective, here's what's happening:
>
> - Our initial quantity is 1 (for all time)
> - This principal earns 100% continuous interest, and after time x
> has earned $\int 1 \, dx = x$. (After 1 unit of time, this is 1)
> - After time x, that interest (x) has earned $\int x \, dx = \frac 12 x^2 $. (After 1 unit of time, this is $\frac 12$)
> - After time x, that interest ($\frac 12 x^2$) has earned $\int \frac 12 x^2 \, dx = \frac 1{3!} x^3 $ (After 1 unit of time, this is \frac 1{3!} = \frac 16)
>
> And so on. Every instant, the entire chain of interest is growing. When learning calculus, you might have repeatedly tried to integrate x just for fun (whatever gets you going). That game is how we end up with e.
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/6laH7.jpg
[2]: https://i.stack.imgur.com/gx2cK.jpg
Chgg Clou
·
2021-02-18T06:36:11Z (almost 4 years ago)