# Riemann Sums

In order to explore Riemann Sums and apply them in real life situations, I did a taxi assignment. Over the weekend, I took a taxi to a friend’s house. On my way there, I recorded the starting odometer and the final odometer to calculate the distance I traveled (about 6.5 kilometers). Furthermore, every 30 seconds I recorded the speedometer reading, and I used this data to estimate the total distance I traveled and compared it to the actual distance I traveled. Using the formula distance = rate x time, I estimated the area under the curve using Riemann Sums and the trapezoidal rule.My left Riemann Sum estimation gave me about 4.266 kilometers, my right Riemann Sum estimation gave me 4.3 kilometers, and my trapezoidal rule estimate which gave me 4.283 kilometers, significantly lower than the 6.5 kilometers I actually traveled. Several times my value was 0 which accounts for the discrepancy because it completely negates the distance I traveled in that 30 second time span.
To determine my left Riemann Sum, I converted 30 seconds into 1/120 hours and multiplied 1/120 hours by all my recorded values except the last one. To determine my right Riemann Sum, I converted 30 seconds into 1/120 hours and multiplied 1/120 hours by all my recorded values except the first one. To determine my trapezoidal rule estimate, I averaged my left and right Riemann Sums. I started calculating midpoint estimation, but I realized in this case the midpoint estimate would equal the trapezoidal estimate because the graph is composed of linear lines.
The average value I estimated was 32.25 kilometers per hour, meaning 32.25 was my average velocity during my taxi ride. My estimated average rate of change was only 8/15 kilometers an hour. This is because the taxi is stopped when I first get in, and towards the end of my ride, the taxi is moving slowly because we were in a neighborhood. My actual rate of change is 0 kph because when I got in the taxi, my velocity was 0, and when I got out my...