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Applications of Integration — Revision Notes

Tamil Nadu Board · Class 12 · Mathematics

Applications of Integration revision notes for Tamil Nadu Board Class 12 Mathematics: 4 topics in quick points.

45 questions25 flashcards10 formulas & key relations5 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 1 & 9.2 — Definite Integral as the Limit of a Sum (Riemann Integral), 3 — Fundamental Theorems and Properties of Definite Integrals, 4 — Bernoulli's Formula for Integration by Parts, 5 — Improper Integrals. Study these first, then practise questions on each for the Tamil Nadu Board Class 12 board exam.
How should I revise Applications of Integration for the Tamil Nadu Board Class 12 board exam?
Learn the core ideas first, then work through the 45 practice questions on Applications of Integration. Revise 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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All Tamil Nadu Board Class 12 Mathematics Revision Notes

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