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Volume by Washers

Volume by Washers
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2016-03-20 09:53
CalculusIntegralGeometric Construction
In collections AP Calculus volume problems
This example demonstrates the volume of a solid of revolution using the washer method, based on AP Exam 2006 Problem 1b. Key mathematical concepts include integration and volume of solids of revolution. GeoGebra commands used include Intersect, Polygon, Rotate, Integral, Sequence, Surface, RectangleSum, etc. Suitable for high school calculus courses.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw a solid of revolution around the x-axis generated by two functions f(x) and g(x) in the interval [a, b]. First, plot the graphs of the two functions, then rotate them around the x-axis to generate the outer and inner surfaces. Next, draw the side surfaces of the outer and inner rings, and finally calculate and display the volume of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.