To find the area under curves, left Riemann sums give us an easy way to break the space into smaller rectangles.

Let’s imagine we have a function called $f(x)$ over a range from $a$ to $b$.

First, we split this range into $n$ smaller sections, which we call sub-intervals. Each of these parts has the same width, which we calculate using this formula:

$\Delta x = \frac{b-a}{n}$

Now, here's the important part: for each sub-interval, we look at the left end to decide how tall our rectangle will be.

For the $i^{th}$ rectangle, the height is determined by the function at that point. We find this by using:

$x_i = a + (i-1) \Delta x$

So, the height of the rectangle is $f(x_i)$. To find the area of each rectangle, we multiply its height by its width, which gives us:

$\text{Area of rectangle} = f(x_i) \Delta x$

In short, the left Riemann sum, which we write as $L_n$, adds up the areas of all these rectangles like this:

$L_n = \sum_{i=1}^{n} f(x_i) \Delta x$

This method helps us estimate the area under the curve.

As we increase $n$ (which means the rectangles become thinner), our estimate becomes more accurate.

For example, if $f(x) = x^2$ from $[0, 2]$, using these left endpoints will give us estimates that improve as we add more rectangles.