# Why does integration find the area under a curve?

$\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx} = {\lim}_{n \to \infty} {\sum}_{i = 1}^{n} f \left(a + i \Delta x\right) \Delta x$,
where $\Delta x = \frac{b - a}{n}$.
If $f \left(x\right) \ge 0$, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of $f \left(x\right)$ above the x-axis.