The area under a curve has no neat plug-in formula, so how can we find it?
The idea is to pack the region with thin rectangles to estimate the area, then slice those rectangles thinner and thinner.
As the rectangles get finer, the estimate creeps toward the true area, and that limit is exactly the definite integral.
Push the n slider higher and watch the Riemann sum converge to the exact value as the error shrinks toward zero.
Slice the Area into Rectangles
📐 Core Idea
①Want the area under a curve → fill with rectangles
②Slice them thinner (n → ∞) → converges to the exact area
③This limit IS the definite integral!
From Riemann Sum to Integral
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As n grows, the Riemann sum converges to the integral — watch the error shrink
Definition
∫ab f(x) dx = limn→∞ Σk=1n f(xk) Δx
Limit of the partition method = area under the curve
🔑 Observations
①n = 2–5: thick rectangles, big error
②n = 20–50: rectangles fill the curve closely
③n → ∞: Riemann sum → integral (exact area)
Fundamental Theorem of Calculus
FTC
∫ab f(x) dx = F(b) - F(a) (with F'(x) = f(x))
Knowing an antiderivative lets you compute the integral directly — no partitions needed!
💡 Why FTC is Powerful
①Direct Riemann limits are complicated
②With FTC: find F(x), compute F(b) − F(a) — done
③Differentiation (slope) and integration (area) are inverses
Worked Example
∫02 x² dx = [x³3]02 = 83 - 0 = 83
Find F(x) = x³/3, then evaluate F(2) − F(0)
Sign of the Integral vs Area
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Area above the x-axis is positive; below is negative — a signed area
Area vs Integral
Area S = ∫ab |f(x)| dx (absolute value!)
Integral is signed; physical area uses |f(x)|
⚠ Watch Out
①Integral value ≠ area (parts below x-axis count as negative)