Area between a curve and the x-axis: For a curve above the x-axis, the definite integral ∫ₐᵇ f(x) dx equals the area between the curve, the x-axis, and the lines x = a and x = b.
The shaded area under a curve between two x-values is the definite integral over that interval.
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Definite = a number: A definite integral (with limits a and b) gives a number — no + C needed.
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Integrate, then F(b) − F(a): Integrate to get F(x) (no + C needed), then substitute the limits: ∫ₐᵇ f dx = F(b) − F(a) — the top limit minus the bottom limit.
IB-style question — evaluate
Evaluate ∫₁³ 2x dx.
Step by step
- Integrate.
- Substitute the limits: F(3) − F(1).
Final answer
∫₁³ 2x dx = 8.
Top minus bottom: Always do F(top) − F(bottom) — reversing the order flips the sign.
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Set the integral equal to the given area: If the area is given and a limit is unknown, write the definite integral with the unknown limit, set it equal to the area, and solve for the limit.
IB-style question — find the limit
The area under y = 2x from x = 0 to x = k is 9.
Find k (k > 0).
Step by step
- Set up the definite integral = 9.
- Solve for the positive k.
Final answer
k = 3.
Keep the sensible root: Solving may give two values; choose the one that fits the context (e.g. positive, or inside the region).