Definite Integration — Class 12 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Definite Integration, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
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Definite Integration in one look
an antiderivative evaluated between two limits, and eight properties that dodge the antiderivative entirely
Definite Integral as a Limit of a Sum
- split [a,b] into n equal strips, width h=(b-a)/n
- sum f at each strip edge, times strip width
- as nto∞, that sum becomes ∫_a^b f(x) dx = lim_hto0 hΣ_r=0^n-1f(a+rh)
- this limit-of-a-sum IS the area; the theorem below skips the limit
Fundamental Theorem of Integral Calculus
- if g is any antiderivative of f, g'(x)=f(x)
∫_a^b f(x) dx = g(b)-g(a)
Evaluate by the fundamental theorem
Q. Evaluate ∫_1^3 x^2 dx.
- antiderivative: g(x) = x^3/3
- g(3)-g(1) = 27/3 - 1/3
Answer: 26/3
Properties I to IV
- Property I: equal limits give zero area
∫_a^a f(x) dx = 0 - Property II: swapping the limits flips the sign
∫_a^b f(x) dx = -∫_b^a f(x) dx - Property III: the integration variable's name never matters
∫_a^b f(x) dx = ∫_a^b f(t) dt - Property IV: split at any c between a and b
∫_a^b f = ∫_a^c f + ∫_c^b f
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What this Definite Integration sheet covers
- Definite Integral as a Limit of a Sum
- Fundamental Theorem of Integral Calculus
- if g is any antiderivative of f, g'(x)=f(x)
- Evaluate by the fundamental theorem
- Properties I to IV
- Property I: equal limits give zero area
- Property II: swapping the limits flips the sign
- Property III: the integration variable's name never matters
- Property IV: split at any c between a and b
- Properties V to VIII
- Property V: reflect x through the midpoint of [a,b]
- Property VI: the special case a=0
- Property VII: reflect around the midpoint of [0,2a]
- Property VIII: even or odd function on a symmetric interval
- Special cases of a definite integral
- Which property actually helps
- Using an odd-function shortcut
- Adding the reflected integral (Property VI)
- Reading the reflection trick
- Definite integral needing substitution first
- Substitution changes the limits too
- Quick reference: all eight properties
- Indefinite vs definite: reading the answer
- Indefinite integral
- Definite integral
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
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