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同心圆环区域面积计算演示

同心圆环区域面积计算演示
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2018-11-22 10:31
微积分积分几何作图
所属专题 GeoGebra 3D & AR: PreCalc & Calculus Resources
①本范例演示如何使用垫圈法（Washer Method）计算同心圆环区域的面积；②涉及的核心数学概念包括积分、函数、几何变换和对称性；③作图要点——从下方『构造摘要』提炼用到了Polygon、Surface、Sequence、Rotate、Textfield等GeoGebra命令/对象；④适合高中及以上学生学习和理解微积分中的积分应用。

GeoGebra 命令及参数

  1. Polygon(A, B, D, C) → q1, a_1, b_1, d, c
  2. Surface(x(B), ((u * y(D)) * cos(v)), ((u * y(D)) * sin(v)), u, 0, 1, v, 0, α) → f
  3. Surface(x(A), ((u * y(D)) * cos(v)), ((u * y(D)) * sin(v)), u, 0, 1, v, 0, α) → g
  4. Sequence(Surface[a + (i * deltax) + (deltax * u), (h(a + ((i + 1) * deltax)) * cos(v)), (h(a + ((i + 1) * deltax)) * sin(v)), u, 0, 1, v, 0, α], i, 0, n - 1, 1) → L_4
  5. Sequence(Polygon[(a + (i * deltax), p(a + ((i + 1) * deltax))), (a + ((i + 1) * deltax), p(a + ((i + 1) * deltax))), (a + ((i + 1) * deltax), h(a + ((i + 1) * deltax))), (a + (i * deltax), h(a + ((i + 1) * deltax)))], i, 0, n - 1) → diffrectangles
  6. Sequence(Surface[a + (i * deltax) + (deltax * u), (p(a + ((i + 1) * deltax)) * cos(v)), (p(a + ((i + 1) * deltax)) * sin(v)), u, 0, 1, v, 0, α], i, 0, n - 1, 1) → l1
  7. Rotate(diffrectangles, α, xAxis) → diffrectangles'
  8. Textfield(h) → InputBox1
  9. Textfield(p) → InputBox2
  10. Textfield(a) → InputBox3
  11. Textfield(b) → InputBox4
  12. Textfield(n) → InputBox5
  13. Sequence(Surface[a + (i * deltax), (cos(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), (sin(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), u, 0, 1, v, 0, α], i, 1, n, 1) → L_2
  14. Sequence(Surface[a + ((i - 1) * deltax), (cos(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), (sin(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), u, 0, 1, v, 0, α], i, 1, n, 1) → L_3