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两曲线间薄板质心的动态演示

两曲线间薄板质心的动态演示
几何画板及绘图内容加载中，请稍等……
2021-08-19 00:48
函数解析几何微积分积分滑块动画
该范例通过GeoGebra动态演示了两条曲线所围薄板的质心位置。核心数学概念是质心（重心）的计算，涉及定积分与积分区间上的面积矩。作图要点包括：使用Intersect求曲线交点，IntegralBetween计算面积，Integral计算面积矩，Point和Midpoint构造关键点，Vector表示质心向量，Segment和Polygon辅助可视化，Sequence生成分割条带，Slider控制参数变化。适合高中或大学低年级的微积分与物理教学，帮助学生直观理解质心公式的几何意义。

GeoGebra 命令及参数

  1. Intersect(f, g) → A, B, C_{1}, D_{1}
  2. Integral((x * h(x)), x(A), x(B)) → b
  3. IntegralBetween(f, g, x(A), x(B)) → a
  4. Point(xAxis) → C
  5. Segment(C, A) → i
  6. Point(xAxis) → D
  7. Segment(D, B) → j
  8. Segment(C, D) → k
  9. Segment(D, C) → l
  10. Sequence((1 + ((k - 1) * d), g(1 + ((k - 1) * d))), k, 1, n) → l1, {(1, 4.2), (1.35, 3.9261250000000008), (1.7, 3.6645000000000003), (2.05, 3.415125), (2.4, 3.178), (2.75, 2.953125), (3.0999999999999996, 2.7405), (3.4499999999999997, 2.5401250000000006), (3.8, 2.3520000000000003), (4.15, 2.176125), (4.5, 2.0125), (4.85, 1.8611250000000004), (5.199999999999999, 1.7220000000000004), (5.55, 1.5951250000000001), (5.8999999999999995, 1.4805000000000001), (6.25, 1.378125), (6.6, 1.288), (6.949999999999999, 1.2101250000000001), (7.3, 1.1445), (7.6499999999999995, 1.0911250000000001), (8, 1.05), (8.35, 1.021125), (8.7, 1.0045), (9.049999999999999, 1.000125), (9.399999999999999, 1.008), (9.75, 1.028125), (10.1, 1.0605), (10.45, 1.105125), (10.799999999999999, 1.162), (11.149999999999999, 1.2311249999999998), (11.5, 1.3125), (11.85, 1.4061249999999998), (12.2, 1.5119999999999998), (12.549999999999999, 1.6301249999999996), (12.899999999999999, 1.7604999999999995), (13.25, 1.9031250000000002), (13.6, 2.058), (13.95, 2.225125), (14.299999999999999, 2.4044999999999996), (14.649999999999999, 2.5961249999999993)}
  11. Sequence((1 + ((k - 1) * d), g(x(l1(2)) + ((k - 1) * d))), k, 1, n) → l4, {(1, 3.9261250000000008), (1.35, 3.6645000000000003), (1.7, 3.415125), (2.05, 3.178), (2.4, 2.953125), (2.75, 2.7405), (3.0999999999999996, 2.5401250000000006), (3.4499999999999997, 2.3520000000000003), (3.8, 2.176125), (4.15, 2.0125), (4.5, 1.8611250000000004), (4.85, 1.7220000000000004), (5.199999999999999, 1.5951250000000003), (5.55, 1.4805), (5.8999999999999995, 1.378125), (6.25, 1.288), (6.6, 1.2101250000000001), (6.949999999999999, 1.1445000000000003), (7.3, 1.091125), (7.6499999999999995, 1.05), (8, 1.021125), (8.35, 1.0045), (8.7, 1.000125), (9.049999999999999, 1.008), (9.399999999999999, 1.028125), (9.75, 1.0605), (10.1, 1.105125), (10.45, 1.162), (10.799999999999999, 1.2311249999999998), (11.149999999999999, 1.3124999999999996), (11.5, 1.4061249999999998), (11.85, 1.5119999999999998), (12.2, 1.6301249999999996), (12.549999999999999, 1.7604999999999995), (12.899999999999999, 1.9031249999999993), (13.25, 2.058), (13.6, 2.225125), (13.95, 2.4044999999999996), (14.299999999999999, 2.5961249999999993), (14.649999999999999, 2.799999999999999)}
  12. Sequence((1 + ((k - 1) * d), f(x(l1(2)) + ((k - 1) * d))), k, 1, n) → l2, {(1, 4.403874999999999), (1.35, 4.5954999999999995), (1.7, 4.774875), (2.05, 4.942), (2.4, 5.096875), (2.75, 5.2395), (3.0999999999999996, 5.3698749999999995), (3.4499999999999997, 5.4879999999999995), (3.8, 5.593875), (4.15, 5.6875), (4.5, 5.7688749999999995), (4.85, 5.838), (5.199999999999999, 5.894875), (5.55, 5.9395), (5.8999999999999995, 5.971875), (6.25, 5.992), (6.6, 5.999875), (6.949999999999999, 5.9955), (7.3, 5.978875), (7.6499999999999995, 5.95), (8, 5.908875), (8.35, 5.8555), (8.7, 5.789875), (9.049999999999999, 5.712000000000001), (9.399999999999999, 5.621875), (9.75, 5.5195), (10.1, 5.4048750000000005), (10.45, 5.2780000000000005), (10.799999999999999, 5.1388750000000005), (11.149999999999999, 4.987500000000001), (11.5, 4.823875), (11.85, 4.648000000000001), (12.2, 4.459875), (12.549999999999999, 4.259500000000001), (12.899999999999999, 4.046875000000001), (13.25, 3.822), (13.6, 3.5848750000000003), (13.95, 3.3355000000000006), (14.299999999999999, 3.073875000000001), (14.649999999999999, 2.800000000000001)}
  13. Sequence((x(l1(2)) + ((k - 1) * d), f(x(l1(2)) + ((k - 1) * d))), k, 1, n) → l3, {(1.35, 4.403874999999999), (1.7000000000000002, 4.5954999999999995), (2.05, 4.774875), (2.4, 4.942), (2.75, 5.096875), (3.1, 5.2395), (3.4499999999999997, 5.3698749999999995), (3.8, 5.4879999999999995), (4.15, 5.593875), (4.5, 5.6875), (4.85, 5.7688749999999995), (5.199999999999999, 5.838), (5.549999999999999, 5.894875), (5.9, 5.9395), (6.25, 5.971875), (6.6, 5.992), (6.949999999999999, 5.999875), (7.299999999999999, 5.9955), (7.65, 5.978875), (8, 5.95), (8.35, 5.908875), (8.7, 5.8555), (9.049999999999999, 5.789875), (9.399999999999999, 5.712000000000001), (9.749999999999998, 5.621875), (10.1, 5.5195), (10.45, 5.4048750000000005), (10.799999999999999, 5.2780000000000005), (11.149999999999999, 5.1388750000000005), (11.499999999999998, 4.987500000000001), (11.85, 4.823875), (12.2, 4.648000000000001), (12.549999999999999, 4.459875), (12.899999999999999, 4.259500000000001), (13.249999999999998, 4.046875000000001), (13.6, 3.822), (13.95, 3.5848750000000003), (14.299999999999999, 3.3355000000000006), (14.649999999999999, 3.073875000000001), (14.999999999999998, 2.800000000000001)}
  14. Polygon(E, F, I, H) → q1, e, f_1, i_1, h_1
  15. Vector(J, K) → u
  16. Vector(K, J) → v
  17. Midpoint(J, K) → L
  18. Vector(M, N) → w
  19. Vector(N, M) → c
  20. Midpoint(M, N) → O
  21. Vector(Q, P) → m
  22. Vector(S, R) → q
  23. Sequence(Polygon[(1 + ((k - 1) * d), f(x(l1(2)) + ((k - 1) * d))), (x(l1(2)) + ((k - 1) * d), f(x(l1(2)) + ((k - 1) * d))), (x(l1(2)) + ((k - 1) * d), g(x(l1(2)) + ((k - 1) * d))), (x(l1(1)) + ((k - 1) * d), g(x(l1(2)) + ((k - 1) * d)))], k, 1, n - 1) → l5, {0.16721249999999888, 0.32585, 0.47591249999999974, 0.6173999999999991, 0.7503124999999988, 0.874649999999999, 0.9904124999999997, 1.0975999999999981, 1.1962125000000006, 1.28625, 1.3677124999999961, 1.440599999999998, 1.5049125000000014, 1.5606500000000025, 1.6078125000000085, 1.6463999999999945, 1.6764124999999996, 1.697849999999999, 1.7107124999999996, 1.7150000000000034, 1.7107124999999996, 1.697849999999999, 1.6764124999999943, 1.6463999999999999, 1.6078125000000014, 1.5606499999999954, 1.5049124999999997, 1.4405999999999963, 1.3677124999999997, 1.2862500000000026, 1.196212499999998, 1.0976, 0.990412500000005, 0.8746500000000008, 0.7503125000000033, 0.6173999999999964, 0.47591249999999974, 0.32584999999999553, 0.167212499999998}

AI 绘图提示语参考

绘制两条在区间内相交的函数曲线，它们围成一个封闭区域。将该区域沿水平方向分割成若干垂直窄条，每条窄条用多边形表示。在x轴上选取两点，分别向两条曲线的交点作垂线，形成一个参考矩形。计算曲线间区域的面积和关于y轴的一次矩，进而确定质心的位置，并用从区域边界指向质心的向量直观显示质心位置。可通过滑块调节分割数量，观察质心随分割加密的逼近过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。