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Volumes of revolution
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Spin a curve through 360° about an axis and it sweeps out a solid. Integration finds its volume.
About the x-axis
V = π∫ab y2 dx
Square y first, then integrate.
y = x from 0 to 3: V = π∫03 x2 dx = πx3<span>303 = 9π
Square y first, then integrate.
y = x from 0 to 3: V = π∫03 x2 dx = πx3<span>303 = 9π
About the y-axis
V = π∫cd x2 dy
Write x2 in terms of y and use y-limits.
y = x2 from y = 0 to y = 2: x2 = y, so V = π∫02 y dy = 2π
Write x2 in terms of y and use y-limits.
y = x2 from y = 0 to y = 2: x2 = y, so V = π∫02 y dy = 2π
Exact answers
Leave answers in terms of π unless asked for a decimal.
y = 2x + 1 from 0 to 1: y2 = 4x2 + 4x + 1
V = π4x3<span>3 + 2x2 + x01 = 133π
y = 2x + 1 from 0 to 1: y2 = 4x2 + 4x + 1
V = π4x3<span>3 + 2x2 + x01 = 133π
The region under y = x2 + 1 from x = 0 to x = 1 is rotated 360° about the x-axis. Find the volume to 3 s.f.
- y2 = x4 + 2x2 + 1
- πx5<span>5 + 2x33 + x01 = π(15 + 23 + 1)
- = 2815π
Answer: 5.86 cubic units
About the x-axis: V = π∫y2 dx. About the y-axis: V = π∫x2 dy. Square before you integrate.
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