Integrals: Area Under the Curve
Integration finds accumulated totals — area, distance, volume. The Fundamental Theorem of Calculus links derivatives and integrals as inverse operations.
Key Concepts
Riemann Sums
Approximate area under f(x) by dividing into n rectangles. Left, Right, or Midpoint sums. As n→∞, the Riemann sum → the definite integral ∫ᵃᵇ f(x) dx.
Antiderivatives
F(x) is an antiderivative of f(x) if F'(x) = f(x). Power rule backwards: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (where n ≠ −1). C is the constant of integration.
Fundamental Theorem of Calculus
∫ᵃᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x). This connects definite integrals (area) to antiderivatives — the key theorem of calculus.
Applications
Distance from velocity: ∫v(t)dt. Area between curves: ∫(f−g)dx. Volume of revolution using disk/washer method. Many physics formulas use integrals.
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Check Your Understanding
The antiderivative of x³ is:
∫₀² 2x dx = ?
The Fundamental Theorem of Calculus says: