Volume of Revolution: GeoGebra Curve Rotation and Definite Integral
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This GeoGebra example demonstrates how a plane curve or line segment rotated around an axis forms a solid of revolution and how its volume can be visualized and computed. It connects coordinate geometry, line segments, Cartesian curves, similarity transformations, and definite integrals. The construction uses the Segment command to create line segments and the CurveCartesian command to draw Cartesian curves, while objects such as k, f, g, a, α, h, b, d, c, i, j, and e represent curves, parameters, angles, heights, bases, or cross-sections. It is suitable for high-school mathematics, introductory calculus, and GeoGebra teaching demonstrations of disk or shell methods for volumes of revolution.
GeoGebra Commands & Parameters
Segment((a, f(a)), (a, g(a))) → bCurveCartesian(t + (h(t) - g(t)) * sin(α) / 5.0, h(t) - (h(t) - g(t)) * cos(α), t, 2.0, 6.0) → dCurveCartesian(t + (h(t) - f(t)) * sin(α) / 5.0, h(t) - (h(t) - f(t)) * cos(α), t, 2.0, 6.0) → cCurveCartesian(a + (h(t) - g(a)) * sin(t) / 5.0, h(t) - (h(t) - g(a)) * cos(t), t, 0.0*°, α) → iCurveCartesian(a + (h(t) - f(a)) * sin(t) / 5.0, h(t) - (h(t) - f(a)) * cos(t), t, 0.0*°, α) → jSegment((a + (h(a) - g(a)) * sin(α) / 5.0, h(a) - (h(a) - g(a)) * cos(α)), (a + (h(a) - f(a)) * sin(α) / 5.0, h(a) - (h(a) - f(a)) * cos(α))) → e
AI Drawing Prompt Reference
Draw the Cartesian curves f(x) and g(x) over the interval from 2 to 6, and a vertical segment at x = a connecting f(a) to g(a). Then rotate the region between these curves and the segment around an axis inclined at angle α to the horizontal, producing a solid of revolution. Show the resulting rotated surfaces d and c, the generating curves i and j, and the segment e linking their endpoints.
The prompt above was compiled from the original drawing commands and is for reference only.