📐 Riemann Sum Explorer

Skill goal: understand how rectangles approximate the area under a curve, and how more rectangles shrink the error.

Observe

Below is the curve f(x) = x² from x = 0 to x = 2, with 10 rectangles drawn using the Left endpoint rule. Notice the gaps between the rectangle tops and the curve.

Manipulate

Predict

Before you change it, what do you think will happen? Example: “If I switch from Left to Right, will the sum go up or down — and why?”

See

True area = ∫₀² x² dx = [x³/3]₀² = 8/3 ≈ 2.667

Explain — Why it works

Each rectangle has width Δx = (2−0)/n and height f(x*), where x* is the left end, right end, or midpoint of the subinterval. Adding all n rectangle areas gives a Riemann sum. As n grows, Δx shrinks and the rectangles hug the curve more tightly, so the sum converges to the true area. For an increasing function like x² on [0, 2], left endpoints always sit below the curve (underestimate) and right endpoints sit above it (overestimate).

Challenge

Keep learning

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