Midpoint Riemann Sums are usually better at estimating the area under a curve than Left and Right Riemann Sums. I found this really interesting when I was studying calculus. Let’s see why this is true!

1. Understanding the Basics:

   • Riemann Sums help us find out how much space is under a curve by splitting it into rectangles.
   • Left and Right Riemann Sums use the left or right points of each section to figure out how tall the rectangles should be.

2. Why Use the Midpoint?:

   • The Midpoint Riemann Sum looks at the height of rectangles right in the middle of each section. This gives a more balanced estimate.
   • By using the midpoint, it can better show how the function behaves across the whole section. So, if the curve is going up or down, the midpoint height usually gives a fairer average than just using the ends.

3. Visual Representation:

   • Imagine a smiley face curve (like $y = x^2$) going from 0 to 1. If you draw rectangles using left or right points, some will be too tall and some too short. But if you use the midpoints, those rectangles fit the curve much better, giving you a more accurate area estimate.

4. Practical Observation:

   • In class, we often practiced finding the area under curves using different methods. Most of the time, the Midpoint Riemann Sum gave numbers that were really close to the actual area compared to the Left or Right Riemann Sums.

In short, even though it depends on the curve and how many sections you use, Midpoint Riemann Sums usually give better area estimates. They consider the average value of each section. So, if you need a method that gives better accuracy, try using the midpoint!