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Volumes of Revolution between Two Quadratics Rotated around y=-3

Volumes of Revolution between Two Quadratics Rotated around y=-3
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2015-07-16 23:15
Solid GeometryCalculusIntegral
In collections BC Calculus Volumes in Calculus Volumes in Calculus
This example demonstrates the calculation of the volume of revolution formed by rotating two quadratic functions around the line y=-3 using the washer method. Key math concepts include volumes of revolution, washer method, and integral calculus. GeoGebra commands used are Sequence, Rotate, RectangleSum, Surface, Polygon, and Integral. Suitable for high school and above students learning calculus volume calculations and geometric visualization.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[(-1 + (k * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), f(-1 + ((k + p) * h))), (-1 + (k * h), f(-1 + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Rotate(Rectangles, (-α), a) → Rectangles'
  3. RectangleSum(A, -1, -1 + (m * h), m, p) → VolWashers
  4. Surface(-1 + (DK * h), -3 + ((u + 3) * cos(β)), ((u + 3) * sin(β)), u, g(-1 + ((DK + p) * h)), f(-1 + ((DK + p) * h)), β, 0, α) → SideLeft1
  5. Surface(-1 + ((DK + 1) * h), -3 + ((u + 3) * cos(β)), ((u + 3) * sin(β)), u, g(-1 + ((DK + p) * h)), f(-1 + ((DK + p) * h)), β, 0, α) → SideRight1
  6. Surface(u, -3 + ((f(-1 + ((DK + p) * h)) + 3) * cos(β)), ((f(-1 + ((DK + p) * h)) + 3) * sin(β)), u, -1 + (DK * h), -1 + ((DK + 1) * h), β, 0, α) → Outer1
  7. Surface(u, -3 + ((g(-1 + ((DK + p) * h)) + 3) * cos(β)), ((g(-1 + ((DK + p) * h)) + 3) * sin(β)), u, -1 + (DK * h), -1 + ((DK + 1) * h), β, 0, α) → Inner1
  8. Polygon((-1 + (DK * h), g(-1 + ((DK + p) * h))), (-1 + ((DK + 1) * h), g(-1 + ((DK + p) * h))), (-1 + ((DK + 1) * h), f(-1 + ((DK + p) * h))), (-1 + (DK * h), f(-1 + ((DK + p) * h)))) → Rectangle1, b, c, d, e
  9. Rotate(Rectangle1, (-α), a) → Rectangle1'
  10. Integral((π * ((f(x) + 3)^(2) - (g(x) + 3)^(2))), -1, 2) → VolInteg
  11. Rotate(f_1, (-α), a) → i
  12. Rotate(g_1, (-α), a) → j
  13. Surface(u, -3 + ((f(u) + 3) * cos(β)), ((f(u) + 3) * sin(β)), u, -1, 2, β, 0, (2 * π)) → Outer
  14. Surface(u, -3 + ((g(u) + 3) * cos(β)), ((g(u) + 3) * sin(β)), u, -1, 2, β, 0, (2 * π)) → Inner
  15. Sequence(Surface[-1 + ((k + u) * h), -3 + ((3 + f(-1 + ((k + p) * h))) * cos(β)), ((3 + f(-1 + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Outers
  16. Sequence(Surface[-1 + ((k + u) * h), -3 + ((3 + g(-1 + ((k + p) * h))) * cos(β)), ((3 + g(-1 + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Inners
  17. Sequence(Surface[-1 + (k * h), -3 + ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * cos(β)), ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → SidesLeft
  18. Sequence(Surface[-1 + ((k + 1) * h), -3 + ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * cos(β)), ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → SidesRight