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Volumes by Washer Example

Volumes by Washer Example
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2016-03-20 09:44
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of the volume of a solid of revolution using the washer method. It involves core mathematical concepts such as integration, function graphs, and volume calculation of rotated bodies. The construction utilizes GeoGebra commands like Intersect, Sequence, Sum, Polygon, Rotate, Integral, Surface, and RectangleSum. Suitable for high school and above mathematics education, illustrating the practical application of integration in volume calculations.

GeoGebra Commands & Parameters

  1. Intersect(g, f, 0, 2) → A, B
  2. Sequence(Cylinder[(a + (k * h), 1, 0), (a + ((k + 1) * h), 1, 0), 1 + f(a + ((k + p) * h))], k, 0, m - 1) → CO
  3. Sequence(Cylinder[(a + (k * h), 1, 0), (a + ((k + 1) * h), 1, 0), 1 + g(a + ((k + p) * h))], k, 0, m - 1) → CI
  4. Sequence(Polygon[(a + (k * h), f(a + ((k + p) * h))), (a + ((k + 1) * h), f(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h))), (a + (k * h), g(a + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  5. Sequence(If[Defined[Volume[Element[CO, k]]], Volume[Element[CO, k]], 0], k, 1, m) → VCO
  6. Sequence(If[Defined[Volume[Element[CI, k]]], Volume[Element[CI, k]], 0], k, 1, m) → VCI
  7. Sum(VCO - VCI) → VC
  8. Polygon((a + (DK * h), g(a + ((DK + p) * h))), (a + ((DK + 1) * h), g(a + ((DK + p) * h))), (a + ((DK + 1) * h), f(a + ((DK + p) * h))), (a + (DK * h), f(a + ((DK + p) * h)))) → Rectangle1, c, d, e, k
  9. Rotate(Rectangle1, (-α), line) → Rectangle1'
  10. Integral((π * ((g(x) - 1)^(2) - (f(x) - 1)^(2))), a, b) → VolInteg
  11. Rotate(f_1, (-α), line) → i
  12. Rotate(g_1, (-α), line) → j
  13. Rotate(Rectangles, (-α), line) → Rectangles'
  14. Surface(u, 1 + ((f(u) - 1) * cos(β)), ((f(u) - 1) * sin(β)), u, a, b, β, 0, (2 * π)) → Inner
  15. Surface(a + (DK * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((DK + p) * h)), g(a + ((DK + p) * h)), β, 0, α) → SideLeft1
  16. Surface(a + ((DK + 1) * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((DK + p) * h)), g(a + ((DK + p) * h)), β, 0, α) → SideRight1
  17. Surface(u, 1 + ((g(u) - 1) * cos(β)), ((g(u) - 1) * sin(β)), u, a, b, β, 0, (2 * π)) → Outer
  18. Surface(u, 1 + ((f(a + ((DK + p) * h)) - 1) * cos(β)), ((f(a + ((DK + p) * h)) - 1) * sin(β)), u, a + (DK * h), a + ((DK + 1) * h), β, 0, α) → Inner1
  19. Surface(u, 1 + ((g(a + ((DK + p) * h)) - 1) * cos(β)), ((g(a + ((DK + p) * h)) - 1) * sin(β)), u, a + (DK * h), a + ((DK + 1) * h), β, 0, α) → Outer1
  20. Sequence(Surface[u, 1 + ((g(a + ((k + p) * h)) - 1) * cos(β)), ((g(a + ((k + p) * h)) - 1) * sin(β)), u, a + (k * h), a + ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  21. Sequence(Surface[u, 1 + ((f(a + ((k + p) * h)) - 1) * cos(β)), ((f(a + ((k + p) * h)) - 1) * sin(β)), u, a + (k * h), a + ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  22. Sequence(Surface[a + (k * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((k + p) * h)), g(a + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  23. Sequence(Surface[a + ((k + 1) * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((k + p) * h)), g(a + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  24. RectangleSum((π * ((ℯ^((2 * x) - x^(2)) - 1)^(2) - 1)), a, a + (m * h), m, p) → VCyls

AI Drawing Prompt Reference

Create a 3D solid of revolution illustrating the washer method for volume calculation. Two function curves enclose a region that rotates around a horizontal axis, forming inner and outer surfaces with multiple thin cylindrical washers between them. The graphic includes: outer surface, inner surface, several semi-transparent washers, a highlighted rectangular cross-section and its rotated position, and an integral curve. Construction order: first draw the two function curves and mark their intersections, then generate a series of cylindrical washers, then show the rotated rectangle and surfaces, and finally display the volume integral formula. Use semi-transparency to reveal internal structure.
The prompt above was compiled from the original drawing commands and is for reference only.