Rotation about the x-axis V=\pi\int_a^b y^2\,dx Each cross-section is a disc of radius y, area πy². Rotation about the y-axis V=\pi\int x^2\,dy Express x in terms of y and integrate over y-limits. Worked example Volume of revolution The region under y = √x from x = 0 to 4 is rotated about the x-axis. Find the volume. V = π ∫₀⁴ y² dx = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx. = π [x²/2]₀⁴ = π (16/2 − 0) = 8π. ✓ V = 8π ≈ 25.1 cubic units.