Rotate curves around axes to generate 3D solids. Calculate volume using disk, washer, and shell methods with animations.
A bowl, a cone, a sphere, a vase - each is a curve spun around an axis, and this visualizer performs exactly that sweep: draw a curve, set the bounds, choose an axis, and watch an animation grow the 3D solid while the volume is computed by slicing it into disks, washers, or shells. One catch dominates every mistake in this topic: the radius gets squared. Volume is pi times the integral of [f(x)]^2, not of f(x), so doubling a curve's height multiplies the volume by four, not two.
A concrete input and expected output from the current implementation.
Input
y = x^2, rotate about the x-axis, x from 0 to 1, disk method
Expected output
A paraboloid bowl rendered in 3D; volume = pi/5 ≈ 0.628 cubic units.
Each slice perpendicular to the x-axis is a disk of radius x^2, so V = pi * integral of (x^2)^2 dx from 0 to 1 = pi * integral of x^4 dx = pi/5. Squaring the radius is the step students skip most often.
The tool slices the region under the curve into thin strips and sums their rotated contributions. Disks and washers are cross-sections perpendicular to the axis of rotation: a disk contributes pi * (radius)^2 * thickness, a washer the difference pi * (R^2 - r^2) * thickness. Shells are cylinders wrapped around the axis, each contributing 2 * pi * radius * height * thickness. Finer slices converge on the exact integral.
Revolving the same curve about different axes makes different solids: y = x^2 on [0,1] about the x-axis has volume pi/5, about the y-axis pi/2. The tool recomputes the radius for the axis you pick - vertical distance to the x-axis, horizontal distance to the y-axis - and offers the matching method: disks and washers cut perpendicular to the axis, shells cut parallel to it.
Compare the axis first: the same curve and bounds about the x-axis and the y-axis are two different solids - y = x^2 on [0,1] is pi/5 one way and pi/2 the other. Then compare the region: washers need an inner and an outer curve, and the inner radius is subtracted after squaring. If the setup matches, the numbers match.
Yes, when they describe the same solid. Each method partitions the same volume into different slices: y = x^2 on [0,1] about the x-axis is pi/5 by disks and also pi/5 by shells taken in y. They can only disagree when the setup describes a different solid - a different axis, a different region, or a radius that was not squared.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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