Volumes of revolution
Point 4 of 5 — Definite Integration, Areas and Volumes of RevolutionQuestion bank: 0 questions (0 with pictures)
04 Volumes of revolution
About the x-axis
- V = π ∫ab y² dx, where y is the height of the curve above the axis of rotation.
- Example: y = x² rotated about the x-axis from x = 0 to x = 1: V = π ∫01 x⁴ dx = π/5.
- Example: y = √x from x = 0 to x = 4: V = π ∫04 x dx = π[ x²/2 ]04 = 8π.
About the y-axis
- V = π ∫cd x² dy; the curve must be written as x = g(y).
- Example: x = y² from y = 0 to y = 2: V = π ∫02 y⁴ dy = 32π/5.
Volume between two curves
- V = π ∫ (R² − r²) dx, where R is the outer radius and r the inner radius.
- Example: the region between y = x and y = x² for 0 ≤ x ≤ 1 rotated about the x-axis: V = π ∫01 (x² − x⁴) dx = π(⅓ − ⅕) = 2π/15.
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