Definite Integration
Maharashtra Board · Class 12 · Mathematics & Statistics
Quick revision notes for Definite Integration — Maharashtra Board Class 12 Mathematics & Statistics. Key concepts, formulas, and definitions for last-minute revision.
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Definite Integral as Limit of Sum
- Definite integral ∫ᵇₐ f(x) dx represents the area under curve y = f(x) between x = a and x = b
- Derived from dividing interval [a,b] into n equal parts and taking limit as n → ∞
- Formula: ∫ᵇₐ f(x) dx = lim(n→∞) Σᵣ₌₁ⁿ h·f(a + rh) where h = (b-a)/n
Fundamental Theorem of Integral Calculus
- If F(x) is antiderivative of f(x), then ∫ᵇₐ f(x) dx = F(b) - F(a)
- Connects differentiation and integration as inverse operations
- Eliminates need for limit calculations in most practical problems
Properties of Definite Integrals - Basic Properties
- Property I: ∫ₐₐ f(x) dx = 0 (integral over zero width interval)
- Property II: ∫ᵇₐ f(x) dx = -∫ₐᵇ f(x) dx (swapping limits changes sign)
- Property III: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt (integration variable doesn't matter)
Advanced Properties - Substitution Properties
- Property V: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(a + b - x) dx (substitution property)
- Property VI: ∫₀ₐ f(x) dx = ∫₀ₐ f(a - x) dx (special case for [0,a])
- Property VII: ∫₀²ₐ f(x) dx = ∫₀ₐ f(x) dx + ∫₀ₐ f(2a - x) dx (interval doubling)
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