Area between a curve and the y-axis
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Flip to reveal answersHow do you find the area between a curve and the y-axis?
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All 8 Flashcards — Area between a curve and the y-axis
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Question
How do you find the area between a curve and the y-axis?
Answer
Integrate x with respect to y: A = ∫ x dy, using the bottom and top y-values as limits.
Question
Why is it ∫ x dy (not ∫ y dx) for the y-axis?
Answer
The strips are horizontal: width x (across to the y-axis) and tiny height dy. Summing x·dy gives the area.
Question
First step before integrating x dy?
Answer
Rearrange y = f(x) into x = g(y) so the integrand is in terms of y.
Question
Rearrange y = ln x for an x dy integral.
Answer
x = eʸ (undo the natural log with the exponential).
Question
Rearrange y = eˣ for an x dy integral.
Answer
x = ln y (take ln of both sides).
Question
What kind of limits does ∫ x dy use?
Answer
y-values — the bottom (c) and top (d) of the region. Convert any given x-limits to y-values first.
Question
Area between y = x² (x ≥ 0) and the y-axis from y = 0 to 4?
Answer
x = √y, A = ∫₀⁴ y^(1/2) dy = [⅔y^(3/2)]₀⁴ = ⅔(8) = 16/3.
Question
y = x² gives x = √y, not x = −√y. Why?
Answer
The region is for x ≥ 0, so we take the positive square root.
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Topic 5.17 hub
Volumes of revolution (HL only)
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