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Applications of Integration

Tamil Nadu Board · Class 12 · Mathematics

Quick revision notes for Applications of Integration — Tamil Nadu Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

45 questions25 flashcards5 concepts

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Key Topics to Revise

1

9.1 & 9.2 — Definite Integral as the Limit of a Sum (Riemann Integral)

  • A definite integral ∫[a to b] f(x)dx is defined as the limit of a Riemann sum when the number of subintervals n → ∞ and the width of each subinterval → 0.
  • The interval [a,b] is divided into n subintervals. In each subinterval [x_{i-1}, x_i], a point ξ_i is chosen. The Riemann sum is Σ f(ξ_i)(x_i - x_{i-1}).
  • Three evaluation rules for equal-width partitions: Left-end rule (ξ_i = x_{i-1}), Right-end rule (ξ_i = x_i), Mid-point rule (ξ_i = midpoint).
2

9.3 — Fundamental Theorems and Properties of Definite Integrals

  • First Fundamental Theorem: If F(x) = ∫[a to x] f(u)du, then d/dx[F(x)] = f(x). Integration and differentiation are inverse operations.
  • Second Fundamental Theorem: ∫[a to b] f(x)dx = F(b) - F(a), where F(x) is any anti-derivative of f(x). Written as [F(x)]_a^b.
  • The arbitrary constant C cancels out when evaluating definite integrals — no need to add C.
3

9.4 — Bernoulli's Formula for Integration by Parts

  • Bernoulli's formula is an extended version of integration by parts, ideal when one function is a polynomial.
  • Let u = polynomial function, v = easily integrable function (like eˣ, sinx, cosx).
  • Notation: u⁽¹⁾ = du/dx, u⁽²⁾ = d²u/dx², ... (successive derivatives); v₍₁₎ = ∫v dx, v₍₂₎ = ∫v₍₁₎dx, ... (successive integrals).
4

9.5 — Improper Integrals

  • Improper integrals arise when the interval of integration is infinite (first kind) or when f(x) becomes unbounded at some point in [a,b] (second kind).
  • Definition: ∫[a to ∞] f(x)dx = lim(t→∞) ∫[a to t] f(x)dx.
  • Definition: ∫[-∞ to a] f(x)dx = lim(t→-∞) ∫[t to a] f(x)dx.

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Full Notes

Key Concepts

A definite integral ∫ₐᵇ f(x)dxFirst Fundamental TheoremKey properties includeImproper integrals have either infinite limitsReduction formulae express integrals of higher

Frequently Asked Questions

What are the important topics in Applications of Integration for Tamil Nadu Board Class 12 Mathematics?
Key topics in Applications of Integration include Applications of Integration — Complete Chapter Overview, Applications of Integration - Complete Chapter Overview, Applications of Integration - Concept Overview. These are the concepts Tamil Nadu Board Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Applications of Integration — Tamil Nadu Board Class 12 Mathematics?
Understand the core concepts first, then work through the 45 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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