AC Riemann Sums

Motivating Questions

• How can a Riemann sum estimate the area between a curve and the horizontal axis?
• Differences among left, right, middle, and random Riemann sums.
• Writing Riemann sums in abbreviated form.

Concepts

Distance and Position Changes

• The area between a velocity function v(t) and the t-axis over a time interval tells the distance traveled.
• If v(t) is sometimes negative, signed areas indicate the change in position.

Estimating Areas with Riemann Sums

• Riemann sums approximate the area under a curve by summing the areas of rectangles.
• Different methods: left, right, midpoint, and random Riemann sums for determining rectangle heights.

Definitions

Sigma Notation

• Sigma notation, using Σ, simplifies expressing sums of large numbers of terms.
• Example: Σ(k=1 to 100, k) = 1 + 2 + ... + 100.

Riemann Sums

• Use subintervals [a, b] divided into n rectangles.
• Left Riemann sum (Ln), right (Rn), and midpoint (Mn) based on rectangle height location.
  • Ln = Σ(f(xi)Δx) where xi is the left endpoint.
  • Rn = Σ(f(xi)Δx) where xi is the right endpoint.
  • Mn = Σ(f(xi)Δx) where xi is the midpoint.

Applications

Negative Function Values

• Riemann sums consider both positive and negative function values.
• Net signed area: difference between positive and negative areas.

Practical Use

• In estimating total distance traveled using velocity functions.
• In theoretical discussions and computational practices.

Activities and Exercises

• Activities involve sketching, calculations of distances using Ln, Rn, and Mn.
• Exercises include evaluating Riemann sums for given functions, estimating distances from data, and writing Riemann sums.

Summary

• Riemann sums are crucial for estimating areas under curves.
• The choice of endpoints or midpoints for determining rectangle heights affects the sum's accuracy.
• Riemann sums can handle negative values, providing a net signed area estimate.

Key Takeaways

• Understanding of Riemann sums is essential in calculus for computing areas and understanding integrals.
• Sigma notation is a powerful tool for expressing sums.
• Different Riemann sums offer flexibility depending on the context and function characteristics.