2007年AP AB卷第1c题：旋转体体积

GeoGebra 命令及参数

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