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Volume by Washers by Mike May (2)

Volume by Washers by Mike May (2)
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2016-03-20 02:59
Solid GeometryCalculusIntegral
This example demonstrates the volume of a solid of revolution using the washer method, based on AP Exam 2010 Problem 4b. Key mathematical concepts include the volume of a solid of revolution and parabolas. GeoGebra commands used include Sequence, Polygon, Rotate, Integral, Surface, Segment, and RectangleSum. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw a diagram illustrating the volume calculation of a solid of revolution using the washer method. Include several rectangles and surfaces formed by rotating parabolas, with integrals to calculate the volume. Start by plotting a sequence of rectangles, then rotate and combine them into washers. Next, draw the inner and outer surfaces of the solid, and finally combine integral lines and surfaces to display the entire solid, showing the volume calculation process.
The prompt above was compiled from the original drawing commands and is for reference only.