Spin a curve around an axis — it traces out a 3D solid. Two methods compute its volume.
Try this
2
volume from rotating y = √x about the x-axis from 0 to b: V = π ∫₀ᵇ x dx = π b²/2 = 6.28
Disk method
Each cross-section perpendicular to the axis is a disk of radius f(x) and area π[f(x)]²; integrate to stack them up.
Washer method
When the solid has a hole, subtract the inner radius's area from the outer.
Try it
Rotate y = x² about the x-axis, 0 ≤ x ≤ 1. Volume?
V = π ∫₀¹ (x²)² dx = π ∫₀¹ x⁴ dx = π[x⁵/5]₀¹ = π/5.
Your turn
Rotate y = 2 (a horizontal line), 0 ≤ x ≤ 3, about the x-axis. What solid, and what volume?
Methods
- Disks/washers — cross-sections perpendicular to the axis.
- Shells — cylindrical shells parallel to the axis.