Practice Flashcards
How do you find the area between a curve and the y-axis?
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All Flashcards in Topic 5.17
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5.17.18 cards
How do you find the area between a curve and the y-axis?
Integrate x with respect to y: A = ∫ x dy, using the bottom and top y-values as limits.
Why is it ∫ x dy (not ∫ y dx) for the y-axis?
The strips are horizontal: width x (across to the y-axis) and tiny height dy. Summing x·dy gives the area.
First step before integrating x dy?
Rearrange y = f(x) into x = g(y) so the integrand is in terms of y.
Rearrange y = ln x for an x dy integral.
x = eʸ (undo the natural log with the exponential).
Rearrange y = eˣ for an x dy integral.
x = ln y (take ln of both sides).
What kind of limits does ∫ x dy use?
y-values — the bottom (c) and top (d) of the region. Convert any given x-limits to y-values first.
Area between y = x² (x ≥ 0) and the y-axis from y = 0 to 4?
x = √y, A = ∫₀⁴ y^(1/2) dy = [⅔y^(3/2)]₀⁴ = ⅔(8) = 16/3.
y = x² gives x = √y, not x = −√y. Why?
The region is for x ≥ 0, so we take the positive square root.
5.17.28 cards
Volume of revolution about the x-axis?
V = π∫y² dx, between the x-limits — discs of radius y, thickness dx.
Volume of revolution about the y-axis?
V = π∫x² dy, between the y-limits — discs of radius x, thickness dy.
Why is the radius squared in the volume formula?
Each slice is a disc; a disc's area is π·radius², so its volume is π·radius²·thickness.
Does 'y²' mean square just the variable or the whole function?
Square the whole function: y² = [f(x)]². E.g. y = x + 1 gives y² = (x + 1)².
Rotating about the y-axis — what must you find first?
x² in terms of y (rearrange the curve), and use y-limits.
Volume between two curves rotated about an axis?
Washers: V = π∫(R² − r²), outer radius² minus inner radius². Never (R − r)².
y = √x rotated about the x-axis, x = 0 to 4, volume?
π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.
Most common lost mark in volume-of-revolution questions?
Forgetting the π (or the dx/dy), or using (R − r)² instead of R² − r².
Topic 5.17 study notes
Full notes & explanations for Volumes of revolution (HL only)
Math AA exam skills
Paper structures, command terms & tips
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