Practice Flashcards
What integral gives the area under a curve y = f(x) above the x-axis from a to b?
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All Flashcards in Topic 5.12
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5.12.18 cards
What integral gives the area under a curve y = f(x) above the x-axis from a to b?
A = ∫ₐᵇ y dx — the definite integral of y between the two x-values.
What integral gives the area between two curves f (top) and g (bottom)?
A = ∫ₐᵇ (f(x) − g(x)) dx, where a and b are the x-values where the curves meet.
How do you find the limits for an 'area between two curves' question?
Solve f(x) = g(x) (use the GDC to find the intersection points); those x-values are the limits a and b.
How do you find the area between a curve and the y-axis?
Rearrange to x = (function of y) and integrate ∫ x dy between two y-values.
Why can a plain definite integral give the wrong area?
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.
In AI, how do you usually evaluate an area integral?
Set up the integral by hand for the marks, then let the GDC evaluate it (a calculator is allowed on every paper).
How do you decide which curve is the 'top' between two curves?
Test one x-value in the interval; the curve with the larger value there is the top.
A flower bed edge is y = 0.5x², 0 ≤ x ≤ 4. What is its area?
∫₀⁴ 0.5x² dx = 32/3 ≈ 10.7 (square units).
5.12.28 cards
Volume of revolution about the x-axis?
V = ∫ₐᵇ π y² dx — discs of radius y along x.
Volume of revolution about the y-axis?
V = ∫꜀ᵈ π x² dy — discs of radius x; rewrite x² in terms of y, use y-limits.
Why π·(radius)² in the integral?
Each thin slice is a disc; a disc's area is π·radius², and the volume sums the discs.
Rotating about the y-axis: what must you change?
Express x² in terms of y AND switch the limits to the y-values.
Volume of y = x² (0≤x≤2) rotated about the y-axis?
x² = y, y: 0→4, V = ∫₀⁴ πy dy = 8π ≈ 25.1.
Volume between two curves rotated about the x-axis?
V = ∫ π(y_outer² − y_inner²) dx — subtract the inner disc.
Common volume-of-revolution slip?
Forgetting to square the radius, dropping the π, or keeping x-limits on a y-axis solid.
How does the GDC help with volumes of revolution?
Type the set-up integral ∫ π(radius)² and let it evaluate — a calculator is allowed on every AI paper.
Topic 5.12 study notes
Full notes & explanations for Areas & volumes of revolution (HL only)
Math AI exam skills
Paper structures, command terms & tips
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