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Volume by Washers Example

Volume by Washers Example
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2016-03-19 20:20
Solid GeometryCalculusIntegral
This example demonstrates how to calculate the volume of a solid of revolution using the washer method. It involves core calculus concepts like integration, washer method, and rotating function graphs. Constructed using GeoGebra commands such as Sequence, Polygon, Rotate, Integral, Surface, and RectangleSum, it is suitable for high school and above students learning about volume calculations in calculus.

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 * ((2 + g(x))^(2) - (2 + f(x))^(2))), 0, 2.3) → VolInteg
  5. Rotate(Rectangles, (-α), a) → Rectangles'
  6. Rotate(g_1, (-α), a) → i
  7. Rotate(f_1, (-α), a) → j
  8. Surface(u, -2 + ((2 + f(((DK + p) * h))) * cos(β)), ((2 + f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  9. Surface(u, -2 + ((2 + g(((DK + p) * h))) * cos(β)), ((2 + g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  10. Surface((DK * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideLeft1
  11. Surface(((DK + 1) * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideRight1
  12. Surface(u, -2 + ((2 + f(u)) * cos(β)), ((2 + f(u)) * sin(β)), u, 0, 2.3, β, 0, (2 * π)) → InnerSurf
  13. Surface(u, -2 + ((2 + g(u)) * cos(β)), ((2 + g(u)) * sin(β)), u, 0, 2.3, β, 0, (2 * π)) → OuterSurf
  14. Sequence(Surface[u, -2 + ((2 + f(((k + p) * h))) * cos(β)), ((2 + f(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  15. Sequence(Surface[u, -2 + ((2 + g(((k + p) * h))) * cos(β)), ((2 + g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  16. Sequence(Surface[(k * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((k + p) * h)), g(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  17. Sequence(Surface[((k + 1) * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((k + 1 + p) * h)), g(((k + 1 + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  18. RectangleSum(A, 0, (m * h), m, p) → WS

AI Drawing Prompt Reference

Draw a volume calculation example using the washer method to demonstrate the calculation of a solid of revolution. The image includes multiple rectangles and rotated function graphs, with integration used to calculate the volume. Start by drawing a series of rectangles, then rotate these rectangles and the function graphs to form the inner and outer surfaces of the solid. Finally, combine all surfaces to create a complete solid of revolution, with an integral line indicating the volume calculation range.
The prompt above was compiled from the original drawing commands and is for reference only.