Rotating y = √x Around x-axis

Calculus 2 · Axiom Academy

EXAMPLE Rotating Around the x-axis Use the disk method to find the volume of the solid generated by rotating about the x-axis from x = 0 to x = 4. Find the volume of the solid of revolution generated by rotating the curve about the x-axis from x = 0 to x = 4 . The Setup: One Representative Disk Each cross-section perpendicular to the x-axis is a circular disk. Its radius is the height of the curve at that point, , so the disk's area is . Adding the disks from x = 0 to x = 4 gives the volume. Spinning about the x-axis sweeps out a smooth, bowl-shaped paraboloid — wider at x = 4 than at x = 0. Each slice is a disk of radius , and they sum to a volume of cubic units. Nice work — you used the disk method to find the volume of a solid of revolution. Here's what carried the problem: Disk method formula: rotating about the x-axis, the volume is , where R(x) is the radius — the distance from the axis to the curve. Identify the radius: for spun about the x-axis, the radius is just the height of the curve, . Square first, then integrate: collapses the integrand to a clean . Evaluate cleanly: cubic units. The shape: the solid is a paraboloid; the volume is measured in cubic units because it fills 3D space. The same recipe handles any curve spun about the x-axis: find the radius, square it, multiply by π, and integrate over your bounds.

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