Area under and between curves
Practice Flashcards
Flip to reveal answersWhat integral gives the area under a curve y = f(x) above the x-axis from a to b?
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All 8 Flashcards — Area under and between curves
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Question
What integral gives the area under a curve y = f(x) above the x-axis from a to b?
Answer
A = ∫ₐᵇ y dx — the definite integral of y between the two x-values.
Question
What integral gives the area between two curves f (top) and g (bottom)?
Answer
A = ∫ₐᵇ (f(x) − g(x)) dx, where a and b are the x-values where the curves meet.
Question
How do you find the limits for an 'area between two curves' question?
Answer
Solve f(x) = g(x) (use the GDC to find the intersection points); those x-values are the limits a and b.
Question
How do you find the area between a curve and the y-axis?
Answer
Rearrange to x = (function of y) and integrate ∫ x dy between two y-values.
Question
Why can a plain definite integral give the wrong area?
Answer
Area below the x-axis is counted as negative, so the integral gives a signed total; split at the roots (or integrate |f(x)|) for true area.
Question
In AI, how do you usually evaluate an area integral?
Answer
Set up the integral by hand for the marks, then let the GDC evaluate it (a calculator is allowed on every paper).
Question
How do you decide which curve is the 'top' between two curves?
Answer
Test one x-value in the interval; the curve with the larger value there is the top.
Question
A flower bed edge is y = 0.5x², 0 ≤ x ≤ 4. What is its area?
Answer
∫₀⁴ 0.5x² dx = 32/3 ≈ 10.7 (square units).
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Topic 5.12 hub
Areas & volumes of revolution (HL only)
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