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2006年AP AB卷第1b题：体积问题

2006年AP AB卷第1b题：体积问题
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2016-06-26 23:59
函数微积分积分
所属专题 Volumes in AP Calculus Volumes in AP Calculus
该范例演示了2006年AP AB卷第1b题中的体积问题。涉及的核心数学概念包括积分和旋转体体积。通过使用GeoGebra命令如Intersect、Polygon、Integral等，可以构造出旋转体的体积。适合高中数学教学和学习微积分的学生。

GeoGebra 命令及参数

  1. Intersect(f, xAxis, (-1.3731216646760573, 0)) → B
  2. Sequence(Polygon[(aa + (k * h), f(aa + ((k + p) * h))), (aa + ((k + 1) * h), f(aa + ((k + p) * h))), (aa + ((k + 1) * h), g(aa + ((k + p) * h))), (aa + (k * h), g(aa + ((k + p) * h)))], k, 0, m - 1) → Rectangles, {0.11989168777938175, 0.32799428683256315, 0.4911547291915701, 0.60700600002159}
  3. Polygon((aa + (DK * h), f(aa + ((DK + p) * h))), (aa + ((DK + 1) * h), f(aa + ((DK + p) * h))), (aa + ((DK + 1) * h), g(aa + ((DK + p) * h))), (aa + (DK * h), g(aa + ((DK + p) * h)))) → Rectangle1, b, c, d, e
  4. Rotate(Rectangle1, (-α), a) → Rectangle1'
  5. Integral((π * ((2 + f(x))^(2) - 4)), aa, 0) → VolInteg
  6. RectangleSum(A, aa, aa + (m * h), m, p) → WS
  7. Rotate(Rectangles, (-α), a) → Rectangles'
  8. Sequence(Surface[aa + ((k + u) * h), -2 + (2 * cos(β)), (2 * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Inners, {(-1.3731216646760573 + ((u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((1 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((2 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((3 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β)))}
  9. Sequence(Surface[aa + ((k + u) * h), -2 + ((2 + f(aa + ((k + p) * h))) * cos(β)), ((2 + f(aa + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Outers, {(-1.3731216646760573 + ((u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((1 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((1 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((1 + 0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((2 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((2 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((2 + 0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((3 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((3 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((3 + 0.5) * 0.22885361077934288))) * sin(β)))}
  10. Rotate(g_1, (-α), a) → i
  11. Rotate(f_1, (-α), a) → j
  12. Surface(u, -2 + ((2 + g(u)) * cos(β)), ((2 + g(u)) * sin(β)), u, aa, 0, β, 0, (2 * π)) → Inner
  13. Surface(aa + ((k + u) * h), -2 + ((2 + g(aa + ((DK + p) * h))) * cos(β)), ((2 + g(aa + ((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  14. Surface(aa + ((k + u) * h), -2 + ((2 + f(aa + ((DK + p) * h))) * cos(β)), ((2 + f(aa + ((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  15. Surface(aa + (DK * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, f(aa + ((DK + p) * h)), g(aa + ((DK + p) * h)), β, 0, α) → SideLeft1
  16. Surface(aa + ((DK + 1) * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, f(aa + ((DK + p) * h)), g(aa + ((DK + p) * h)), β, 0, α) → SideRight1
  17. Surface(u, -2 + ((2 + f(u)) * cos(β)), ((2 + f(u)) * sin(β)), u, aa, 0, β, 0, (2 * π)) → Outer
  18. Sequence(Surface[aa + (k * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, 0, f(aa + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft, {(-1.3731216646760573 + (0 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (1 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (2 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (3 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)))}
  19. Sequence(Surface[aa + ((k + 1) * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, 0, f(aa + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight, {(-1.3731216646760573 + ((1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((1 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((2 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((3 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)))}
  20. Segment((aa, 0), (0, 0)) → l
  21. Segment((0, 0), (0, f(0))) → q
  22. Rotate(q, (-α), a) → q'