What happened to the $\Delta x$?
In a county the population density per quadratic kilometer is assessed to vary according to $f=\frac{8000}{x\sqrt{x}}$, where $x$ is the distance to the county's center. How many people live in the annulus from $x=1$ to $x=4$?
Solution:
The area of the annulus can be said to equal $2\pi x\Delta x$ (draw a figure), so therefore the amount of people who live in the annulus from $x=1$ to $x=4$ is equal to $\int _1^4dx\frac{8000}{\sqrt{x}x}2\pi x$. But why is the $\Delta x$ not used in the calculation of the integral?
We write $\Delta x$ to represent a change in the value of $x$. We often consider splitting up a region into pieces of size $\Delta x$, for example splitting an annulus into smaller annuli each with thickness $\Delta x$.
But multiplying population times area only works if population is constant. So we want to think of splitting the big annulus into very small annuli, each with infinitesimally small thickness (we are essentially thinking of $\Delta x \to 0$). When we do this we usually replace $\Delta x$ with $dx$. In this way, we replace $$\sum_{i=1}^{n} f(x_{i}) 2 \pi x_{i} \Delta x$$ where $\Delta x = \frac{4-1}{n}$ and $x_{i}$ is some point between $1 + (i-1) \Delta x$ and $1+i \Delta x$ with the limit as $n \to \infty$ (so as $\Delta x \to 0$) which we write as $$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i}) 2 \pi x_{i} \Delta x = \int_{1}^{4} f(x)2\pi x dx.$$