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Look Inside: Geometric and Calculus Integration

Look Inside: Geometric and Calculus Integration
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2019-04-29 02:48
Solid GeometryCalculusGeometric Construction
In collections SPATIAL VISUALISATION TASKS
This example demonstrates the integration of geometric shapes and calculus concepts, highlighting the connection between solid geometry and integral calculations. Key mathematical concepts include geometric construction, fundamental theorem of calculus, and parametric equations. Construction involves using GeoGebra commands such as Intersect, Cube, Translate to build polygons and 3D shapes. Suitable for high school students or college freshmen to understand cross-disciplinary applications of geometry and calculus.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → k, D, stenaCDHG, stenaEFGH, hranaAD, hranaAB, hranaBC, hranaCD, hranaAE, hranaDH, hranaEH, hranaEF, E, hranaBF, hranaFG, hranaCG, hranaGH, F, G, H, stenaABCD, stenaADHE, stenaABFE, stenaBCGF
  5. Intersect(hranaAB, hranaAD) → I
  6. Intersect(hranaAB, hranaBC) → J
  7. Vector(I, J) → u
  8. Intersect(hranaAD, hranaCD) → K
  9. Vector(I, K) → v
  10. Intersect(zAxis, hranaEH) → L
  11. Vector(I, L) → w
  12. Intersect(hranaCG, hranaBC) → M
  13. Intersect(hranaFG, hranaGH) → N
  14. Point(zAxis) → O
  15. Point(yAxis) → P
  16. Vector(I, P) → l
  17. Vector(J, M) → p
  18. Translate(k, Vector[(-2 * v) - (2 * u) - (2 * w) + (e * l)]) → U1
  19. Translate(k, Vector[(-v) - (2 * u) - (2 * w) + (e * l)]) → U2
  20. Translate(k, Vector[(-2 * u) - (2 * w) + (e * l)]) → U3
  21. Translate(k, Vector[v - (2 * u) - (2 * w) + (e * p)]) → U4
  22. Translate(k, Vector[(2 * v) - (2 * u) - (2 * w) + (e * p)]) → U5
  23. Translate(k, Vector[(-2 * v) - (2 * u) - w + (e * l)]) → V1
  24. Translate(k, Vector[(-v) - (2 * u) - w + (e * l)]) → V2
  25. Translate(k, Vector[(2 * v) - (2 * u) - w + (e * p)]) → V5
  26. Translate(k, Vector[(-2 * u) - w + (e * p)]) → V3
  27. Translate(k, Vector[(-2 * u) + v - w + (e * p)]) → V4
  28. Translate(k, Vector[(-2 * u) - (2 * v) + (e * l)]) → X1
  29. Translate(k, Vector[(-2 * u) - v + (e * l)]) → X2
  30. Translate(k, Vector[(-2 * u) + (e * l)]) → X3
  31. Translate(k, Vector[(-2 * u) + v + (e * p)]) → X4
  32. Translate(k, Vector[(-2 * u) + (2 * v) + (e * p)]) → X5
  33. Translate(k, Vector[(-2 * u) + w - (2 * v) + (e * l)]) → Y1
  34. Translate(k, Vector[(-2 * u) + w - v + (e * l)]) → Y2
  35. Translate(k, Vector[(-2 * u) + w + (e * p)]) → Y3
  36. Translate(k, Vector[(-2 * u) + w + v + (e * p)]) → Y4
  37. Translate(k, Vector[(-2 * u) + w + (2 * v) + (e * p)]) → Y5
  38. Translate(k, Vector[(-2 * u) + (2 * w) - (2 * v) + (e * l)]) → Z1
  39. Translate(k, Vector[(-2 * u) + (2 * w) - v + (e * l)]) → Z2
  40. Translate(k, Vector[(-2 * u) + (2 * w) + (e * l)]) → Z3
  41. Translate(k, Vector[(-2 * u) + (2 * w) + v + (e * p)]) → Z4
  42. Translate(k, Vector[(-2 * u) + (2 * w) + (2 * v) + (e * p)]) → Z5
  43. Translate(k, Vector[(-u) - (2 * w) - (2 * v) + (d * l)]) → P1
  44. Translate(k, Vector[(-u) - (2 * w) - v + (d * l)]) → P2
  45. Translate(k, Vector[(-u) - (2 * w) + (d * p)]) → P3
  46. Translate(k, Vector[(-u) - (2 * w) + v + (d * p)]) → P4
  47. Translate(k, Vector[(-u) - (2 * w) + (2 * v) + (d * p)]) → P5
  48. Translate(k, Vector[(-u) - w - (2 * v) + (d * l)]) → Q1
  49. Translate(k, Vector[(-u) - w - v + (d * l)]) → Q2
  50. Translate(k, Vector[(-u) - w + (d * l)]) → Q3
  51. Translate(k, Vector[(-u) - w + v + (d * p)]) → Q4
  52. Translate(k, Vector[(-u) - w + (2 * v) + (d * p)]) → Q5
  53. Translate(k, Vector[(-u) - (2 * v) + (d * l)]) → R1
  54. Translate(k, Vector[(-u) - v + (d * l)]) → R2
  55. Translate(k, Vector[(-u) + (d * p)]) → R3
  56. Translate(k, Vector[(-u) + v + (d * p)]) → R4
  57. Translate(k, Vector[(-u) + (2 * v) + (d * p)]) → R5
  58. Translate(k, Vector[(-u) - (2 * v) + w + (d * l)]) → S1
  59. Translate(k, Vector[(-u) - v + w + (d * l)]) → S2
  60. Translate(k, Vector[(-u) + w + (d * l)]) → S3
  61. Translate(k, Vector[(-u) + v + w + (d * p)]) → S4
  62. Translate(k, Vector[(-u) + (2 * v) + w + (d * p)]) → S5
  63. Translate(k, Vector[(-u) - (2 * v) + (2 * w) + (d * l)]) → T1
  64. Translate(k, Vector[(-u) - v + (2 * w) + (d * l)]) → T2
  65. Translate(k, Vector[(-u) + (2 * w) + (d * p)]) → T3
  66. Translate(k, Vector[(-u) + (2 * w) + v + (d * p)]) → T4
  67. Translate(k, Vector[(-u) + (2 * w) + (2 * v) + (d * p)]) → T5
  68. Translate(k, Vector[(-2 * w) - (2 * v) + (c * l)]) → K11
  69. Translate(k, Vector[(-2 * w) - v + (c * l)]) → K22
  70. Translate(k, Vector[(-2 * w) + (c * l)]) → K33
  71. Translate(k, Vector[(-2 * w) + v + (c * p)]) → K44
  72. Translate(k, Vector[(-2 * w) + (2 * v) + (c * p)]) → K55
  73. Translate(k, Vector[(-w) - (2 * v) + (c * l)]) → L11
  74. Translate(k, Vector[(-w) - v + (c * l)]) → L22
  75. Translate(k, Vector[(-w) + v + (c * p)]) → L44
  76. Translate(k, Vector[(-w) + (c * p)]) → L33
  77. Translate(k, Vector[(-w) + (2 * v) + (c * p)]) → L55
  78. Translate(k, Vector[(-2 * v) + (c * l)]) → M11
  79. Translate(k, Vector[(-v) + (c * l)]) → M22
  80. Translate(k, Vector[(c * l)]) → M33
  81. Translate(k, Vector[v + (c * p)]) → M44
  82. Translate(k, Vector[(2 * v) + (c * p)]) → M55
  83. Translate(k, Vector[w - (2 * v) + (c * l)]) → N11
  84. Translate(k, Vector[w - v + (c * l)]) → N22
  85. Translate(k, Vector[w + (c * p)]) → N33
  86. Translate(k, Vector[w + v + (c * p)]) → N44
  87. Translate(k, Vector[w + (2 * v) + (c * p)]) → N55
  88. Translate(k, Vector[(2 * w) - (2 * v) + (c * l)]) → O11
  89. Translate(k, Vector[(2 * w) - v + (c * l)]) → O22
  90. Translate(k, Vector[(2 * w) + (c * l)]) → O33
  91. Translate(k, Vector[(2 * w) + v + (c * p)]) → O44
  92. Translate(k, Vector[(2 * w) + (2 * v) + (c * p)]) → O55
  93. Translate(k, Vector[u - (2 * w) - (2 * v) + (b * l)]) → F11
  94. Translate(k, Vector[u - (2 * w) - v + (b * l)]) → F22
  95. Translate(k, Vector[u - (2 * w) + (b * p)]) → F33
  96. Translate(k, Vector[u - (2 * w) + v + (b * p)]) → F44
  97. Translate(k, Vector[u - (2 * w) + (2 * v) + (b * p)]) → F55
  98. Translate(k, Vector[u - w - (2 * v) + (b * l)]) → G11
  99. Translate(k, Vector[u - w - v + (b * l)]) → G22
  100. Translate(k, Vector[u - w + (b * l)]) → G33
  101. Translate(k, Vector[u - w + v + (b * p)]) → G44
  102. Translate(k, Vector[u - w + (2 * v) + (b * p)]) → G55
  103. Translate(k, Vector[u - (2 * v) + (b * l)]) → H11
  104. Translate(k, Vector[u - v + (b * l)]) → H22
  105. Translate(k, Vector[u + (b * p)]) → H33
  106. Translate(k, Vector[u + v + (b * p)]) → H44
  107. Translate(k, Vector[u + (2 * v) + (b * p)]) → H55
  108. Translate(k, Vector[u + w - (2 * v) + (b * l)]) → I11
  109. Translate(k, Vector[u + w - v + (b * l)]) → I22
  110. Translate(k, Vector[u + w + (b * l)]) → I33
  111. Translate(k, Vector[u + w + v + (b * p)]) → I44
  112. Translate(k, Vector[u + w + (2 * v) + (b * p)]) → I55
  113. Translate(k, Vector[u + (2 * w) - (2 * v) + (b * l)]) → J11
  114. Translate(k, Vector[u + (2 * w) - v + (b * l)]) → J22
  115. Translate(k, Vector[u + (2 * w) + (b * p)]) → J33
  116. Translate(k, Vector[u + (2 * w) + v + (b * p)]) → J44
  117. Translate(k, Vector[u + (2 * w) + (2 * v) + (b * p)]) → J55
  118. Translate(k, Vector[(2 * u) - (2 * w) - (2 * v) + (a * l)]) → A11
  119. Translate(k, Vector[(2 * u) - (2 * w) - v + (a * l)]) → A22
  120. Translate(k, Vector[(2 * u) - (2 * w) + (a * l)]) → A33
  121. Translate(k, Vector[(2 * u) - (2 * w) + v + (a * p)]) → A44
  122. Translate(k, Vector[(2 * u) - (2 * w) + (2 * v) + (a * p)]) → A55
  123. Translate(k, Vector[(2 * u) - w - (2 * v) + (a * l)]) → B11
  124. Translate(k, Vector[(2 * u) - w - v + (a * l)]) → B22
  125. Translate(k, Vector[(2 * u) - w + (a * p)]) → B33
  126. Translate(k, Vector[(2 * u) - w + v + (a * p)]) → B44
  127. Translate(k, Vector[(2 * u) - w + (2 * v) + (a * p)]) → B55
  128. Translate(k, Vector[(2 * u) - (2 * v) + (a * l)]) → C11
  129. Translate(k, Vector[(2 * u) - v + (a * l)]) → C22
  130. Translate(k, Vector[(2 * u) + (a * l)]) → C33
  131. Translate(k, Vector[(2 * u) + v + (a * p)]) → C44
  132. Translate(k, Vector[(2 * u) + (2 * v) + (a * p)]) → C55
  133. Translate(k, Vector[w + (2 * u) - (2 * v) + (a * l)]) → D11
  134. Translate(k, Vector[w + (2 * u) - v + (a * l)]) → D22
  135. Translate(k, Vector[w + (2 * u) + (a * p)]) → D33
  136. Translate(k, Vector[w + (2 * u) + v + (a * p)]) → D44
  137. Translate(k, Vector[w + (2 * u) + (2 * v) + (a * p)]) → D55
  138. Translate(k, Vector[(2 * w) + (2 * u) - (2 * v) + (a * l)]) → E11
  139. Translate(k, Vector[(2 * w) + (2 * u) - v + (a * l)]) → E22
  140. Translate(k, Vector[(2 * w) + (2 * u) + (a * l)]) → E33
  141. Translate(k, Vector[(2 * w) + (2 * u) + v + (a * p)]) → E44
  142. Translate(k, Vector[(2 * w) + (2 * u) + (2 * v) + (a * p)]) → E55
  143. Point(yAxis) → Q
  144. Polygon(Q, R, 4) → A0, f_6, g_6, h_6, i_6, S, T
  145. Point(f_6) → U
  146. Polygon(Q, U, 4) → A1, j_6, k_6, l_6, m_6, V, W
  147. Point(f_6) → Z
  148. Polygon(U, Z, 4) → A2, n_6, p_6, q_6, r_6, A_1, B_1
  149. Point(f_6) → C_1
  150. Polygon(Z, C_1, 4) → A3, s_6, t_6, a_6, b_6, D_1, E_1
  151. Point(f_6) → F_1
  152. Polygon(C_1, F_1, 4) → A4, c_6, d_6, e_6, f_7, G_1, H_1
  153. Polygon(F_1, R, 4) → A5, g_7, h_7, i_7, j_7, I_1, J_1
  154. Polygon(W, V, 4) → B1, k_7, l_7, m_7, n_7, K_1, L_1
  155. Polygon(V, A_1, 4) → B2, p_7, q_7, r_7, s_7, M_1, N_1
  156. Polygon(A_1, D_1, 4) → B3, t_7, a_7, b_7, c_7, O_1, P_1
  157. Polygon(D_1, G_1, 4) → B4, d_7, e_7, f_8, g_8, Q_1, R_1
  158. Polygon(G_1, I_1, 4) → B5, h_8, i_8, j_8, k_8, S_1, T_1
  159. Polygon(L_1, K_1, 4) → C1, l_8, m_8, n_8, p_8, U_1, V_1
  160. Polygon(K_1, M_1, 4) → C2, q_8, r_8, s_8, t_8, W_1, Z_1
  161. Polygon(M_1, O_1, 4) → C3, a_8, b_8, c_8, d_8, A_2, B_2
  162. Polygon(O_1, Q_1, 4) → C4, e_8, f_9, g_9, h_9, C_2, D_2
  163. Polygon(Q_1, S_1, 4) → C5, i_9, j_9, k_9, l_9, E_2, F_2
  164. Polygon(V_1, U_1, 4) → D1, m_9, n_9, p_9, q_9, G_2, H_2
  165. Polygon(U_1, W_1, 4) → D2, r_9, s_9, t_9, a_9, I_2, J_2
  166. Polygon(W_1, A_2, 4) → D3, b_9, c_9, d_9, e_9, K_2, L_2
  167. Polygon(A_2, C_2, 4) → D4, f_{10}, g_{10}, h_{10}, i_{10}, M_2, N_2
  168. Polygon(C_2, E_2, 4) → D5, j_{10}, k_{10}, l_{10}, m_{10}, O_2, P_2
  169. Polygon(H_2, G_2, 4) → E1, n_{10}, p_{10}, q_{10}, r_{10}, Q_2, R_2
  170. Polygon(G_2, I_2, 4) → E2, s_{10}, t_{10}, a_{10}, b_{10}, S_2, T_2
  171. Polygon(I_2, K_2, 4) → E3, c_{10}, d_{10}, e_{10}, f_{11}, U_2, V_2
  172. Polygon(K_2, M_2, 4) → E4, g_{11}, h_{11}, i_{11}, j_{11}, W_2, Z_2
  173. Polygon(M_2, O_2, 4) → E5, k_{11}, l_{11}, m_{11}, n_{11}, A_3, B_3
  174. Intersect(f_6, i_6) → C_3
  175. Point(f_6) → D_3
  176. Polygon(C_3, D_3, 4) → F1, p_{11}, q_{11}, r_{11}, s_{11}, E_3, F_3
  177. Polygon(D_3, G_3, 4) → F2, t_{11}, a_{11}, b_{11}, c_{11}, H_3, I_3
  178. Point(f_6) → J_3
  179. Polygon(G_3, J_3, 4) → F3, d_{11}, e_{11}, f_{12}, g_{12}, K_3, L_3
  180. Point(f_6) → M_3
  181. Polygon(J_3, M_3, 4) → F4, h_{12}, i_{12}, j_{12}, k_{12}, N_3, O_3
  182. Intersect(f_6, g_6) → P_3
  183. Polygon(M_3, P_3, 4) → F5, l_{12}, m_{12}, n_{12}, p_{12}, Q_3, R_3
  184. Polygon(F_3, E_3, 4) → G1, q_{12}, r_{12}, s_{12}, t_{12}, S_3, T_3
  185. Polygon(E_3, H_3, 4) → G2, a_{12}, b_{12}, c_{12}, d_{12}, U_3, V_3
  186. Polygon(H_3, K_3, 4) → G3, e_{12}, f_{13}, g_{13}, h_{13}, W_3, Z_3
  187. Polygon(K_3, N_3, 4) → G4, i_{13}, j_{13}, k_{13}, l_{13}, A_4, B_4
  188. Polygon(N_3, Q_3, 4) → G5, m_{13}, n_{13}, p_{13}, q_{13}, C_4, D_4
  189. Polygon(T_3, S_3, 4) → H1, r_{13}, s_{13}, t_{13}, a_{13}, E_4, F_4
  190. Polygon(S_3, U_3, 4) → H2, b_{13}, c_{13}, d_{13}, e_{13}, G_4, H_4
  191. Polygon(U_3, W_3, 4) → H3, f_{14}, g_{14}, h_{14}, i_{14}, I_4, J_4
  192. Polygon(W_3, A_4, 4) → H4, j_{14}, k_{14}, l_{14}, m_{14}, K_4, L_4
  193. Polygon(A_4, C_4, 4) → H5, n_{14}, p_{14}, q_{14}, r_{14}, M_4, N_4
  194. Polygon(F_4, E_4, 4) → I1, s_{14}, t_{14}, a_{14}, b_{14}, O_4, P_4
  195. Polygon(E_4, G_4, 4) → I2, c_{14}, d_{14}, e_{14}, f_{15}, Q_4, R_4
  196. Polygon(G_4, I_4, 4) → I3, g_{15}, h_{15}, i_{15}, j_{15}, S_4, T_4
  197. Polygon(I_4, K_4, 4) → I4, k_{15}, l_{15}, m_{15}, n_{15}, U_4, V_4
  198. Polygon(K_4, M_4, 4) → I5, p_{15}, q_{15}, r_{15}, s_{15}, W_4, Z_4
  199. Polygon(P_4, O_4, 4) → J1, t_{15}, a_{15}, b_{15}, c_{15}, A_5, B_5
  200. Polygon(O_4, Q_4, 4) → J2, d_{15}, e_{15}, f_{16}, g_{16}, C_5, D_5
  201. Polygon(Q_4, S_4, 4) → J3, h_{16}, i_{16}, j_{16}, k_{16}, E_5, F_5
  202. Polygon(S_4, U_4, 4) → J4, l_{16}, m_{16}, n_{16}, p_{16}, G_5, H_5
  203. Polygon(U_4, W_4, 4) → J5, q_{16}, r_{16}, s_{16}, F1_2, I_5, J_5
  204. SelectedIndex(l1) → o_6
  205. Intersect(f_6, i_6) → K_5
  206. Point(f_6) → L_5
  207. Polygon(K_5, L_5, 4) → K1, t_{16}, a_{16}, b_{16}, c_{16}, M_5, N_5
  208. Point(f_6) → O_5
  209. Polygon(L_5, O_5, 4) → K2, d_{16}, e_{16}, f_{17}, g_{17}, P_5, Q_5
  210. Point(f_6) → R_5
  211. Polygon(O_5, R_5, 4) → K3, h_{17}, i_{17}, j_{17}, k_{17}, S_5, T_5
  212. Point(f_6) → U_5
  213. Polygon(R_5, U_5, 4) → K4, l_{17}, m_{17}, n_{17}, p_{17}, V_5, W_5
  214. Intersect(f_6, g_6) → Z_5
  215. Polygon(U_5, Z_5, 4) → K5, q_{17}, r_{17}, s_{17}, t_{17}, A_6, B_6
  216. Polygon(N_5, M_5, 4) → L1, a_{17}, b_{17}, c_{17}, d_{17}, C_6, D_6
  217. Polygon(M_5, P_5, 4) → L2, e_{17}, f_{18}, g_{18}, h_{18}, E_6, F_6
  218. Polygon(P_5, S_5, 4) → L3, i_{18}, j_{18}, k_{18}, l_{18}, G_6, H_6
  219. Polygon(S_5, V_5, 4) → L4, m_{18}, n_{18}, p_{18}, q_{18}, I_6, J_6
  220. Polygon(V_5, A_6, 4) → L5, r_{18}, s_{18}, t_{18}, a_{18}, K_6, L_6
  221. Polygon(D_6, C_6, 4) → M1, b_{18}, c_{18}, d_{18}, e_{18}, M_6, N_6
  222. Polygon(C_6, E_6, 4) → M2, f_{19}, g_{19}, h_{19}, i_{19}, O_6, P_6
  223. Polygon(E_6, G_6, 4) → M3, j_{19}, k_{19}, l_{19}, m_{19}, Q_6, R_6
  224. Polygon(G_6, I_6, 4) → M4, n_{19}, p_{19}, q_{19}, r_{19}, S_6, T_6
  225. Polygon(I_6, K_6, 4) → M5, s_{19}, t_{19}, a_{19}, b_{19}, U_6, V_6
  226. Polygon(N_6, M_6, 4) → N1, c_{19}, d_{19}, e_{19}, f_{20}, W_6, Z_6
  227. Polygon(M_6, O_6, 4) → N2, g_{20}, h_{20}, i_{20}, j_{20}, A_7, B_7
  228. Polygon(O_6, Q_6, 4) → N3, k_{20}, l_{20}, m_{20}, n_{20}, C_7, D_7
  229. Polygon(Q_6, S_6, 4) → N4, p_{20}, q_{20}, r_{20}, s_{20}, E_7, F_7
  230. Polygon(S_6, U_6, 4) → N5, t_{20}, a_{20}, b_{20}, c_{20}, G_7, H_7
  231. Polygon(Z_6, W_6, 4) → O1, d_{20}, e_{20}, f_{21}, g_{21}, I_7, J_7
  232. Polygon(W_6, A_7, 4) → O2, h_{21}, i_{21}, j_{21}, k_{21}, K_7, L_7
  233. Polygon(A_7, C_7, 4) → O3, l_{21}, m_{21}, n_{21}, p_{21}, M_7, N_7
  234. Polygon(C_7, E_7, 4) → O4, q_{21}, r_{21}, s_{21}, t_{21}, O_7, P_7
  235. Polygon(E_7, G_7, 4) → O5, a_{21}, b_{21}, c_{21}, d_{21}, Q_7, R_7