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Volume of Solids with Square Cross Sections

Volume of Solids with Square Cross Sections
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2016-06-27 00:17
PolygonSolid GeometryCalculusIntegralLocus
In collections Volumes in AP Calculus Volumes in AP Calculus
This GeoGebra example demonstrates the calculation of a solid's volume when cross sections perpendicular to an axis are squares. It illustrates the core calculus concept of using definite integrals to accumulate area of known cross sections to find total volume. Key mathematical ideas include integral accumulation, geometric interpretation of volume, and properties of squares and polygons. The construction utilizes GeoGebra commands such as Intersect, Polygon, Segment, Locus, Integral, If, and objects like Square, A, B, C, D, along with sequences and locus tracing. It is suitable for AP Calculus BC, college-level integral calculus, and students learning volume integration techniques.

GeoGebra Commands & Parameters

  1. Intersect(f, g, (1.0328318883641743, 2.401108092689728)) → L
  2. Polygon(B, A, C, D) → Square, c, d, e, i
  3. Segment(A, B) → b
  4. Locus(D, s) → loc1
  5. Integral((ht(x))^(2), a, 2) → Integ
  6. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  7. Locus(C, s) → loc2
  8. Angle(B, A, C) → α
  9. If(togS ≟ 0, "Show Surface", "Hide Surface") → ShowSurface
  10. Surface(u, f(u), (v * (f(u) - g(u))), u, xl, 2, v, 0, 1) → Back
  11. Surface(u, (v * g(u)) + ((1 - v) * f(u)), 0, u, xl, 2, v, 0, 1) → Base
  12. Surface(u, g(u), (v * (f(u) - g(u))), u, xl, 2, v, 0, 1) → Front
  13. Surface(u, (v * g(u)) + ((1 - v) * f(u)), f(u) - g(u), u, xl, 2, v, 0, 1) → Top
  14. Sequence(Polygon[(a + (k * h), g(a + ((k + p) * h)), 0), (a + (k * h), f(a + ((k + p) * h)), 0), (a + (k * h), f(a + ((k + p) * h)), ht(a + ((k + p) * h))), (a + (k * h), g(a + ((k + p) * h)), ht(a + ((k + p) * h)))], k, 0, m - 1) → SlicesLeft, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  15. Sequence(Area[Element[SlicesLeft, k]], k, 1, m) → Areas, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  16. Sequence(Surface[a + u, f(a + ((k + p) * h)), (v * ht(a + ((k + p) * h))), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesBack, {(1.0328318883641743 + u, f(1.0328318883641743 + ((0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))))}
  17. Sequence(Polygon[(a + (k * h), g(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h))), (a + ((k + 1) * h), f(a + ((k + p) * h))), (a + (k * h), f(a + ((k + p) * h)))], k, 0, m - 1) → SlicesBase, {0.029648810900253064, 0.08745431199746978, 0.13841758867205822, 0.17630270854615393, 0.1944463541914181}
  18. Sequence(Surface[a + u, g(a + ((k + p) * h)), (v * ht(a + ((k + p) * h))), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesFront, {(1.0328318883641743 + u, g(1.0328318883641743 + ((0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))))}
  19. Sequence(Polygon[(a + ((k + 1) * h), g(a + ((k + p) * h)), 0), (a + ((k + 1) * h), f(a + ((k + p) * h)), 0), (a + ((k + 1) * h), f(a + ((k + p) * h)), ht(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h)), ht(a + ((k + p) * h)))], k, 0, m - 1) → SlicesRight, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  20. Sequence(Surface[a + u, (v * g(a + ((k + p) * h))) + ((1 - v) * f(a + ((k + p) * h))), ht(a + ((k + p) * h)), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesTop, {(1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))}