Area under x² from 0 to 2: antiderivative gives exact answer
∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3 ≈ 2.667. The antiderivative F(x) = x³/3 gives the exact area without approximation.
The two parts of the Fundamental Theorem of Calculus
Part 1: d/dx [∫ₐˣ f(t)dt] = f(x)
Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a) where F'(x) = f(x)
Part 1 says differentiation undoes integrasion. Part 2 says to evaluate an integral, find an antiderivative dan subtract the endpoints.
Riemann sum approximating area: FTC replaces this with one calculation
A Riemann sum with 8 rectangles gives ≈ 0.273. The exact answer is 8/3 ≈ 2.667. The Fundamental Theorem gives exact results with no rectangles needed.
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