Volumes of revolution
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Flip to reveal answersVolume of revolution about the x-axis?
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Question
Volume of revolution about the x-axis?
Answer
V = π∫y² dx, between the x-limits — discs of radius y, thickness dx.
Question
Volume of revolution about the y-axis?
Answer
V = π∫x² dy, between the y-limits — discs of radius x, thickness dy.
Question
Why is the radius squared in the volume formula?
Answer
Each slice is a disc; a disc's area is π·radius², so its volume is π·radius²·thickness.
Question
Does 'y²' mean square just the variable or the whole function?
Answer
Square the whole function: y² = [f(x)]². E.g. y = x + 1 gives y² = (x + 1)².
Question
Rotating about the y-axis — what must you find first?
Answer
x² in terms of y (rearrange the curve), and use y-limits.
Question
Volume between two curves rotated about an axis?
Answer
Washers: V = π∫(R² − r²), outer radius² minus inner radius². Never (R − r)².
Question
y = √x rotated about the x-axis, x = 0 to 4, volume?
Answer
π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.
Question
Most common lost mark in volume-of-revolution questions?
Answer
Forgetting the π (or the dx/dy), or using (R − r)² instead of R² − r².
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Topic 5.17 hub
Volumes of revolution (HL only)
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