Zero width, swapping limits, and splitting: Useful facts: ∫ₐᵃ f = 0 (zero-width); ∫ₐᵇ f = −∫_b^a f (swapping limits flips the sign); and ∫ₐᵇ f + ∫_b^c f = ∫ₐ^c f (you can split or join an interval). Constants and sums also pull through.
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.
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.
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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.