CalcRef

Applications of Integrals · Volume

Washer Method

V=π∫ab([R(x)]2−[r(x)]2)dxV = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) dx

Volume of revolution when there is a gap between the curve and the axis. R(x) is the outer radius and r(x) is the inner radius.

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find the volume when the region between y = x² and y = x is revolved about the x-axis.
  1. Intersection: x² = x → x = 0, 1. On [0,1]: R = x, r = x²
  2. V = π ∫₀¹ (x² - x⁴) dx = π[x³/3 - x⁵/5]₀¹ = π(1/3 - 1/5) = 2π/15

Answer: 2π/15 ≈ 0.4189

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