Left vs Right Riemann Sums

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About This MicroSim

This visualization allows students to compare left and right Riemann sum approximations:

• Blue rectangles show the left Riemann sum (using left endpoints)
• Orange rectangles show the right Riemann sum (using right endpoints)
• The green curve shows the actual function
• The info panel displays numerical values and error analysis

Iframe Code

<iframe src="https://dmccreary.github.io/calculus/sims/left-right-riemann/main.html" height="552px" scrolling="no" style="width: 100%;"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

1. Distinguish between left and right Riemann sums visually and computationally
2. Analyze when left sums overestimate vs underestimate (and vice versa for right sums)
3. Explain the relationship between function behavior (increasing/decreasing) and approximation error

Activities

1. Discovery (5 min): Observe both approximations on an increasing function. Which is larger?
2. Analysis (10 min): Switch to a decreasing function. Does the relationship flip?
3. Convergence (5 min): Increase n and watch both approximations converge to the true value

Key Insights

• For increasing functions: Left sum underestimates, Right sum overestimates
• For decreasing functions: Left sum overestimates, Right sum underestimates
• As n increases, both sums converge to the true integral

References

• Riemann Sums - Khan Academy