Zero width, swapping limits, and splitting: Useful facts: ∫ₐᵃ f = 0 (zero-width); ∫ₐᵇ f = −∫_ba f (swapping limits flips the sign); and ∫ₐᵇ f + ∫_bc f = ∫ₐc f (you can split or join an interval).
Constants and sums also pull through.
A definite integral is the signed area between the curve and the x-axis from a to b.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
IB-style question — use the split property
Given ∫₀³ f(x) dx = 7 and ∫₀⁵ f(x) dx = 12, find ∫₃⁵ f(x) dx.
Step by step
- Use ∫₀³ + ∫₃⁵ = ∫₀⁵.
- Subtract.
Final answer
∫₃⁵ f(x) dx = 5.
Split at a shared limit: When two integrals share a limit, you can add or subtract them to get the piece you need.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Antiderivative, then F(b) − F(a): Integrate the trig/exponential, then substitute the limits.
Use ∫cos x dx = sin x, ∫sin x dx = −cos x, ∫eˣ dx = eˣ (and the linear-inside ÷ a versions).
IB-style question — trig & exp
Evaluate ∫₀π/2 cos x dx and ∫₀¹ e2x dx.
Step by step
- First integral.
- Second (∫e2x = ½e2x).
Final answer
∫₀π/2 cos x dx = 1; ∫₀¹ e2x dx = ½(e² − 1) ≈ 3.19.
Radians for trig limits: Trig integrals use radian limits (e.g. π/2).
Keep your GDC in radian mode for these.
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
Integrate a rate to get the accumulated change: If you know a rate of change, the definite integral of the rate gives the total change: the amount at time b is (amount at a) + ∫ₐᵇ (rate) dt.
On Paper 2 use the GDC's numerical integral.
IB-style question — accumulated change
A population changes at rate dP/dt = −80e−0.2t per year.
If P = 2000 at t = 0, find P when t = 10.
Step by step
- Total change = ∫₀¹⁰ (rate) dt (GDC).
- Add to the starting amount.
Final answer
P(10) ≈ 1654.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.