1. Calculate the Width ($\Delta x$): $(b - a) / n$. (This is your rectangle base).
2. Pick your Points):
- LRAM: Use $x_0, x_1, ... x_{n-1}$.
- RRAM: Use $x_1, x_2, ... x_n$.
- MRAM: Use the values in the exact middle of each sub-interval.
3. Find the Heights: Plug your points into $f(x)$.
4. Sum it Up: $\text{Area} \approx \Delta x \cdot (\text{Sum of Heights})$.
Using the function $f(x) = x^2 + 1$. We are looking at the area from $x = 0$ to $x = 4$.
The Big Picture (n=1): Calculate the LRAM estimate using only ONE rectangle for the entire interval $[0, 4]$.
The Overshoot (n=1): Calculate the RRAM estimate using only ONE rectangle for the entire interval $[0, 4]$.
Look at the difference between 4 and 68! Which one is a better estimate of the true area? Why is a single box such a poor tool for measuring a curve? How does this remind you of making a "Quick Judgment" on a whole person based on one moment?
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Use the same function $f(x) = x^2 + 1$ from $x=0$ to $x=4$, but now divide it into 4 sub-intervals.
LRAM (Left-hand): Find the sum using heights at $x = 0, 1, 2, 3$.
RRAM (Right-hand): Find the sum using heights at $x = 1, 2, 3, 4$.
MRAM (Midpoint): Find the sum using heights at $x = 0.5, 1.5, 2.5, 3.5$.
On the grid below, sketch the curve $y = \sqrt{x}$ from $x=0$ to $x=9$. Draw 3 rectangles using RRAM ($n=3$, width=3).
Shade the "Overhang"—the part of the rectangles that is above the curve.
If a function is perfectly flat, $f(x) = 10$.
Does it matter if we use LRAM, RRAM, or MRAM?
Does it matter if we use 1 rectangle or 1,000,000?
Why is the "Estimate" for a constant always exactly equal to the "Truth"?
Objective: Explain Riemann Sums to a younger student using a handful of coins and a piece of paper.
The Activity:
1. Draw a large curved shape on the paper.
2. Try to fill the shape with pennies.
3. "See the gaps between the circles? That's our 'Error'."
4. Now try to fill it with grains of sand.
5. "The smaller the pieces, the better they fit the shape."
The Lesson: "Integration is like using infinite grains of sand to fill our life perfectly."
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