When we talk about Riemann sums, they help us figure out the area under a curve. There are two main types: left Riemann sums and right Riemann sums. The difference between them is about where we measure the height of the rectangles we draw to estimate that area.

1. Left Riemann Sum

• This method uses the left side of each small section (or subinterval) to find out how tall each rectangle should be.
• For instance, if we’re looking at a section from point $a$ to point $b$ and we divide it into $n$ equal parts, we find the height of each rectangle by using the value of the function at the left side. We call this $f(x_i)$, where $x_i$ is the left endpoint.

2. Right Riemann Sum

• On the other hand, the right Riemann sum uses the right side of each small section to determine the height.
• So, using the same section from $a$ to $b$, the height is found with $f(x_{i+1})$, where $x_{i+1}$ is the right endpoint.

Why Is This Important?

• Both methods help us get an estimate of the area, but they can give different answers depending on what kind of curve we are looking at.
• If the curve is going up, the right sum might give us a number that's too high, while the left sum could be too low. If the curve is going down, it works the other way around.

The good news is, if you use more sections (making $n$ bigger), your estimate will get better, no matter which method you choose.

Understanding these ideas is really important because they set the stage for what you’ll learn in calculus later!