Definite Integrals
Kerala Board · Class 12 · Mathematics
Quick revision notes for Definite Integrals — Kerala Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Definition and Geometric Interpretation
- Definite integral as limit of sum: ∫[a to b] f(x)dx = lim(n→∞) h[f(a) + f(a+h) + ... + f(a+(n-1)h)] where h = (b-a)/n
- Represents area under curve y = f(x) from x = a to x = b
- Upper limit 'b' and lower limit 'a' are fixed values
Properties of Definite Integrals
- Property 1: ∫[a to b] f(x)dx = -∫[b to a] f(x)dx (reversing limits changes sign)
- Property 2: ∫[a to c] f(x)dx = ∫[a to b] f(x)dx + ∫[b to c] f(x)dx (additive property)
- Property 3: ∫[a to b] f(x)dx = ∫[a to b] f(a+b-x)dx (substitution property)
Evaluation by Substitution Method
- Step 1: Choose appropriate substitution u = g(x)
- Step 2: Find du = g'(x)dx
- Step 3: Change limits: when x = a, u = g(a); when x = b, u = g(b)
Applications - Area Under Curves
- Area under y = f(x) from x = a to x = b: A = ∫[a to b] f(x)dx (when f(x) ≥ 0)
- Area between two curves: A = ∫[a to b] |f(x) - g(x)|dx
- When f(x) > g(x): A = ∫[a to b] [f(x) - g(x)]dx
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Sources & Official References
- Kerala Board of Public Examinations — keralapareekshabhavan.in
- National Education Policy 2020 — education.gov.in
Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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