Definite Integration and Area Under a Curve
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All 8 Flashcards — Definite Integration and Area Under a Curve
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Question
What is a definite integral?
Answer
An integral with limits [a, b] that gives a specific number — the signed area between the curve and the x-axis from x = a to x = b.
💡 Hint
Unlike indefinite integrals, no +C is needed.
Question
State the Fundamental Theorem of Calculus.
Answer
∫[a to b] f(x) dx = F(b) − F(a), where F is any antiderivative of f.
💡 Hint
Evaluate F at b, then subtract F at a.
Question
Evaluate ∫[1 to 3] 2x dx.
Answer
F(x) = x². F(3) − F(1) = 9 − 1 = 8.
💡 Hint
Integrate to get F(x), then apply limits.
Question
If f(x) < 0 on [a, b], what does the definite integral give?
Answer
A negative number. The integral gives signed area — negative when the curve is below the x-axis. For total area, take the absolute value.
💡 Hint
Below x-axis = negative integral.
Question
How do you find the area between two curves y = f(x) and y = g(x)?
Answer
1) Find intersections: solve f(x) = g(x) to get limits a and b. 2) Identify the top function. 3) Integrate [f(x) − g(x)] from a to b.
💡 Hint
Always: top minus bottom.
Question
Find the area under y = x² + 1 from x = 0 to x = 2.
Answer
∫[0 to 2] (x²+1) dx = [x³/3 + x] from 0 to 2 = (8/3 + 2) − 0 = 14/3 ≈ 4.67 square units.
💡 Hint
Integrate then evaluate F(2) − F(0).
Question
On IB Paper 2, how can you evaluate definite integrals?
Answer
Use your GDC. But always write the integral notation first (e.g., ∫[a to b] f(x) dx = ...). Marks are given for the setup, not just the answer.
💡 Hint
GDC gives the number; marks need the setup.
Question
Area between y = x and y = x² from x = 0 to x = 1.
Answer
∫[0 to 1] (x − x²) dx = [x²/2 − x³/3] from 0 to 1 = 1/2 − 1/3 = 1/6 square units.
💡 Hint
y=x is above y=x² on [0,1]. Integrate top − bottom.
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