Volume via Triangular Cross Sections

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2015-11-02 06:02

In collections Calculus: Volumes of Solids Calculus: Volumes of Solids

This example is designed to help visualize objects constructed from triangular cross sections. Users can set the type of triangle, two functions, limits, axis of rotation, and number of sections. Adjusting these parameters allows for a visual understanding of volume changes under different conditions. Key mathematical concepts include integral calculus and solid geometry. The construction utilizes GeoGebra commands and objects such as Intersect, Textfield, Min, Max, IntegralBetween, Segment, Plane, SelectedIndex, and Polygon. It is suitable for high school and above students learning volume calculation methods, particularly in calculus and solid geometry education.

GeoGebra Commands & Parameters

Intersect(g, l_c, (1, 1)) → P_{gc1}
Intersect(f, l_c, (1, 1)) → P_{fc1}
Textfield(a) → inputBox_a
Textfield(b) → inputBox_b
Min(abs(rad_f), abs(rad_g)) → rad_i
Max(abs(rad_f), abs(rad_g)) → rad_o
Intersect(f, l_b, (5, 5)) → P_{fb}
Intersect(g, l_b, (5, 0.2)) → P_{gb}
Intersect(f, l_a, (1, 1)) → P_{fa}
Intersect(g, l_a, (1, 1)) → P_{ga}
IntegralBetween(f, g, Min[x(P_{fa}), x(P_{ga})], Max[x(P_{fb}), x(P_{gb})]) → R
Segment((0, y_{rotation}), (0, If[abs(rad_f) ≥ abs(rad_g), y(P_{fc1}), y(P_{gc1})])) → r_o
Segment((0, y_{rotation}), (0, If[abs(rad_f) < abs(rad_g), y(P_{fc1}), y(P_{gc1})])) → r_i
Plane((x_i, y_{rotation}, rad_o), x = x_i) → pln3d
Textfield(n) → InputBox1
SelectedIndex(opt) → optIndex
Polygon(P_{fc1}, P_{gc1}, opt_{v1}(SelectedIndex[opt])) → crosssec_1, seg_{11}, seg_{12}, seg_{13}
Polygon(P_{fc2}, P_{gc2}, opt_{v2}(SelectedIndex[opt])) → crosssec_2, seg_{21}, seg_{22}, seg_{23}
Polygon(P_{fc1}, P_{gc1}, opt_{v3}(SelectedIndex[opt])) → crosssec_3, seg_{31}, seg_{32}, seg_{33}
SelectedIndex(lstTraceOpt) → lstTOIndex
Textfield(tf) → InputBox_f
Textfield(tg) → InputBox_g