Properties of Definite Integral Demo

Properties of Definite Integral Demo
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2015-02-01 23:26
This example demonstrates the fundamental properties of definite integrals and their applications. It involves core math concepts like definite integral definition, interval division, function graphing, and integral calculation. Constructed using GeoGebra commands such as Integral, Text, and AttachCopyToView, with objects like f, a, b, c, d. Suitable for high school and above students to explore definite integral properties or for teacher demonstrations.

GeoGebra Commands & Parameters

  1. Integral(f, a, b) → d
  2. Text(d) → text1
  3. AttachCopyToView(PolyLine[(7.440000000000004, -1.2600000000000038), (7.3800000000000034, -1.2400000000000038), (7.300000000000004, -1.2400000000000038), (7.220000000000004, -1.2400000000000038), (7.140000000000004, -1.2600000000000038), (7.080000000000004, -1.3000000000000038), (7.0000000000000036, -1.3400000000000039), (6.9600000000000035, -1.400000000000004), (6.940000000000004, -1.460000000000004), (6.900000000000004, -1.520000000000004), (6.8800000000000034, -1.580000000000004), (6.860000000000004, -1.6400000000000041), (6.860000000000004, -1.720000000000004), (6.860000000000004, -1.800000000000004), (6.860000000000004, -1.8800000000000041), (6.860000000000004, -1.9600000000000042), (6.860000000000004, -2.040000000000004), (6.8800000000000034, -2.120000000000004), (6.8800000000000034, -2.220000000000004), (6.900000000000004, -2.2800000000000042), (6.9200000000000035, -2.3600000000000043), (6.940000000000004, -2.4400000000000044), (6.940000000000004, -2.5200000000000045), (6.940000000000004, -2.6200000000000045), (6.940000000000004, -2.7000000000000046), (6.940000000000004, -2.7800000000000042), (6.9200000000000035, -2.8400000000000043), (6.900000000000004, -2.9000000000000044), (6.860000000000004, -2.9600000000000044), (6.840000000000003, -3.0200000000000045), (6.780000000000004, -3.1200000000000045), (6.740000000000004, -3.1800000000000046), (6.720000000000003, -3.2400000000000047), (6.680000000000003, -3.3000000000000047), (6.620000000000004, -3.3600000000000048), (6.560000000000003, -3.400000000000005), (6.5000000000000036, -3.420000000000005), (6.440000000000004, -3.440000000000005), (6.360000000000004, -3.460000000000005), (6.220000000000003, -3.460000000000005), (6.120000000000004, -3.460000000000005), (6.040000000000004, -3.460000000000005), (5.980000000000003, -3.400000000000005), (5.9600000000000035, -3.3400000000000047), (5.940000000000003, -3.2800000000000047), (NaN, NaN), (7.020000000000004, -3.540000000000005), (6.940000000000004, -3.540000000000005), (6.860000000000004, -3.540000000000005), (6.800000000000003, -3.560000000000005), (6.740000000000004, -3.6000000000000045), (6.740000000000004, -3.6800000000000046), (6.720000000000003, -3.7600000000000047), (6.720000000000003, -3.8400000000000047), (6.740000000000004, -3.900000000000005), (6.800000000000003, -3.960000000000005), (6.860000000000004, -3.980000000000005), (6.940000000000004, -3.980000000000005), (7.020000000000004, -3.980000000000005), (7.080000000000004, -3.960000000000005), (7.100000000000004, -3.900000000000005), (7.100000000000004, -3.8200000000000047), (7.100000000000004, -3.7200000000000046), (7.100000000000004, -3.6400000000000046), (7.120000000000004, -3.7000000000000046), (7.140000000000004, -3.7600000000000047), (7.160000000000004, -3.8200000000000047), (7.180000000000004, -3.900000000000005), (7.200000000000004, -3.960000000000005), (7.200000000000004, -4.0400000000000045), (7.240000000000004, -4.100000000000005), (NaN, NaN), 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(8.080000000000004, -1.500000000000004), (8.060000000000004, -1.560000000000004), (8.000000000000004, -1.600000000000004), (7.940000000000004, -1.620000000000004), (7.860000000000004, -1.620000000000004), (NaN, NaN), (8.760000000000005, -2.200000000000004), (8.640000000000004, -2.200000000000004), (8.560000000000004, -2.200000000000004), (8.460000000000004, -2.220000000000004), (8.380000000000004, -2.240000000000004), (8.300000000000004, -2.2600000000000042), (8.240000000000004, -2.3000000000000043), (8.180000000000003, -2.3600000000000043), (8.140000000000004, -2.4400000000000044), (8.100000000000005, -2.5000000000000044), (8.100000000000005, -2.5800000000000045), (8.100000000000005, -2.6600000000000046), (8.100000000000005, -2.7600000000000042), (8.100000000000005, -2.8400000000000043), (8.100000000000005, -2.9200000000000044), (8.120000000000005, -2.9800000000000044), (8.120000000000005, -3.0800000000000045), (8.120000000000005, -3.1600000000000046), (8.080000000000004, 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-3.2800000000000047), (11.460000000000004, -3.2800000000000047), (11.520000000000005, -3.3200000000000047), (11.580000000000004, -3.3600000000000048), (11.620000000000005, -3.420000000000005), (11.680000000000005, -3.460000000000005), (11.720000000000004, -3.520000000000005), (11.740000000000004, -3.5800000000000045), (11.780000000000005, -3.6400000000000046), true], 1, (-4.300000000000001, -6.1600000000000055), (12.020000000000005, 6.299999999999999), (0, 623), (816, 0)) → stroke1
  4. Integral(f, a, c) → g
  5. Integral(f, c, b) → h
  6. Text(g) → text2
  7. Text(h) → text3
  8. AttachCopyToView(PolyLine[(-1.9199999999999995, -2.5800000000000045), (-1.9999999999999996, -2.5800000000000045), (-2.0799999999999996, -2.5800000000000045), (-2.1599999999999997, -2.5800000000000045), (-2.2399999999999998, -2.5800000000000045), (-2.3, -2.6000000000000045), (-2.38, -2.6400000000000046), (-2.44, -2.6800000000000046), (-2.5, -2.720000000000004), (-2.5599999999999996, -2.7800000000000042), (-2.5999999999999996, -2.8400000000000043), (-2.6199999999999997, -2.9000000000000044), (-2.6399999999999997, -2.9600000000000044), (-2.6399999999999997, -3.0400000000000045), (-2.6399999999999997, -3.1200000000000045), (-2.6399999999999997, -3.2000000000000046), (-2.6199999999999997, -3.2600000000000047), (-2.5999999999999996, -3.3400000000000047), (-2.5799999999999996, -3.420000000000005), (-2.5799999999999996, -3.500000000000005), (-2.5599999999999996, -3.560000000000005), (-2.5199999999999996, -3.6200000000000045), (-2.5, -3.6800000000000046), (-2.48, -3.7400000000000047), (-2.48, 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(3.6000000000000023, -2.9400000000000044), (3.6000000000000023, -2.8600000000000043), (3.6000000000000023, -2.7800000000000042), (3.6000000000000023, -2.7000000000000046), (3.6600000000000024, -2.6800000000000046), (3.7400000000000024, -2.6800000000000046), (3.8200000000000025, -2.6800000000000046), (3.8800000000000026, -2.7000000000000046), (3.9400000000000026, -2.720000000000004), (4.000000000000003, -2.740000000000004), (4.040000000000003, -2.8000000000000043), (4.040000000000003, -2.8800000000000043), (4.020000000000002, -2.9600000000000044), (4.000000000000003, -3.0200000000000045), (3.9400000000000026, -3.0400000000000045), (3.8800000000000026, -3.0800000000000045), (3.8200000000000025, -3.1000000000000045), (3.7600000000000025, -3.1200000000000045), (3.6800000000000024, -3.1200000000000045), (NaN, NaN), (4.8000000000000025, -4.000000000000004), (4.900000000000003, -4.000000000000004), (4.980000000000003, -4.000000000000004), (5.060000000000003, -4.000000000000004), (5.140000000000003, -4.000000000000004), (5.200000000000003, -4.020000000000005), (5.300000000000003, -4.020000000000005), (5.3800000000000034, -4.020000000000005), (NaN, NaN), (4.880000000000003, -4.400000000000005), (5.000000000000003, -4.400000000000005), (5.080000000000003, -4.400000000000005), (5.160000000000003, -4.400000000000005), (5.240000000000003, -4.400000000000005), (5.320000000000003, -4.400000000000005), (5.400000000000003, -4.400000000000005), (NaN, NaN), (7.860000000000004, -2.7000000000000046), (7.620000000000004, -2.6600000000000046), (7.520000000000004, -2.6600000000000046), (7.3800000000000034, -2.6600000000000046), (7.260000000000004, -2.6600000000000046), (7.200000000000004, -2.720000000000004), (7.140000000000004, -2.8000000000000043), (7.060000000000004, -2.9200000000000044), (7.0000000000000036, -3.0200000000000045), (6.940000000000004, -3.1400000000000046), (6.900000000000004, -3.2000000000000046), (6.860000000000004, -3.2800000000000047), (6.800000000000003, -3.3200000000000047), (6.760000000000003, -3.420000000000005), (6.680000000000003, -3.520000000000005), (6.640000000000003, -3.6000000000000045), (6.600000000000003, -3.6800000000000046), (6.580000000000004, -3.7600000000000047), (6.560000000000003, -3.8200000000000047), (6.520000000000003, -3.900000000000005), (6.5000000000000036, -3.960000000000005), (6.4600000000000035, -4.020000000000005), (6.4600000000000035, -4.1200000000000045), (6.4600000000000035, -4.200000000000005), (6.4600000000000035, -4.320000000000005), (6.4600000000000035, -4.400000000000005), (6.4600000000000035, -4.480000000000005), (6.4600000000000035, -4.560000000000005), (6.5000000000000036, -4.620000000000005), (6.5000000000000036, -4.7000000000000055), (6.5000000000000036, -4.800000000000005), (6.5000000000000036, -4.880000000000005), (6.5000000000000036, -4.980000000000005), (6.5000000000000036, -5.060000000000005), (6.440000000000004, -5.120000000000005), (6.400000000000004, -5.2000000000000055), (6.360000000000004, -5.260000000000005), (6.300000000000003, -5.300000000000005), (6.260000000000003, -5.360000000000006), (6.180000000000003, -5.400000000000006), (6.120000000000004, -5.440000000000006), (6.060000000000003, -5.460000000000005), (6.0000000000000036, -5.480000000000006), (5.9200000000000035, -5.500000000000005), (5.840000000000003, -5.500000000000005), (5.760000000000003, -5.500000000000005), (5.680000000000003, -5.460000000000005), (5.620000000000003, -5.420000000000005), (NaN, NaN), (7.080000000000004, -5.380000000000005), (6.9200000000000035, -5.380000000000005), (6.860000000000004, -5.400000000000006), (6.840000000000003, -5.460000000000005), (6.820000000000004, -5.520000000000006), (6.800000000000003, -5.580000000000005), (6.780000000000004, -5.640000000000006), (6.800000000000003, -5.7000000000000055), (6.8800000000000034, -5.7000000000000055), (6.9600000000000035, -5.7000000000000055), (7.060000000000004, -5.7000000000000055), (7.160000000000004, -5.7000000000000055), (7.220000000000004, -5.680000000000006), (7.220000000000004, -5.600000000000006), (7.220000000000004, -5.520000000000006), (7.200000000000004, -5.460000000000005), (7.240000000000004, -5.520000000000006), (7.300000000000004, -5.580000000000005), (7.360000000000004, -5.620000000000005), (7.400000000000004, -5.680000000000006), (NaN, NaN), (8.260000000000003, -2.4800000000000044), (8.260000000000003, -2.6000000000000045), (8.260000000000003, -2.7000000000000046), (8.260000000000003, -2.8200000000000043), (8.260000000000003, -2.9000000000000044), (8.260000000000003, -2.9800000000000044), (8.260000000000003, -3.0600000000000045), (8.260000000000003, -3.1600000000000046), (8.260000000000003, -3.2400000000000047), (8.240000000000004, -3.1800000000000046), (8.240000000000004, -3.1000000000000045), (8.240000000000004, -3.0200000000000045), (8.260000000000003, -2.9600000000000044), (8.260000000000003, -2.8800000000000043), (8.300000000000004, -2.8200000000000043), (8.340000000000003, -2.740000000000004), (8.340000000000003, -2.6600000000000046), (8.400000000000004, -2.6200000000000045), (8.500000000000004, -2.6200000000000045), (8.560000000000004, -2.6400000000000046), (8.620000000000005, -2.6800000000000046), (8.680000000000005, -2.720000000000004), (8.700000000000005, -2.7800000000000042), (8.720000000000004, -2.8400000000000043), (8.720000000000004, -2.9200000000000044), (8.720000000000004, -3.0200000000000045), (8.700000000000005, -3.0800000000000045), (8.660000000000004, -3.1400000000000046), (8.580000000000004, -3.2000000000000046), (8.500000000000004, -3.2400000000000047), (8.420000000000003, -3.2600000000000047), (8.340000000000003, -3.2600000000000047), (8.280000000000005, -3.2800000000000047), (8.200000000000005, -3.2800000000000047), (8.180000000000003, -3.2200000000000046), (8.160000000000004, -3.1400000000000046), (8.160000000000004, -3.0600000000000045), (8.160000000000004, -2.9800000000000044), (8.160000000000004, -2.9000000000000044), (8.140000000000004, -2.8400000000000043), (8.140000000000004, -2.7600000000000042), (8.140000000000004, -2.6800000000000046), (8.140000000000004, -2.6000000000000045), (8.200000000000005, -2.5600000000000045), (8.220000000000004, -2.5000000000000044), (8.240000000000004, -2.4200000000000044), (8.260000000000003, -2.3600000000000043), (8.260000000000003, -2.2800000000000042), (8.260000000000003, -2.200000000000004), (8.220000000000004, -2.140000000000004), true], 1, (-4.300000000000001, -6.1600000000000055), (12.020000000000005, 6.299999999999999), (0, 623), (816, 0)) → stroke2

AI Drawing Prompt Reference

Draw the graph of the function f and display the area under the curve from a to b. Also, separately show the areas under the curve from a to c and from c to b, and label their results. Finally, draw all these areas and the function graph in the same coordinate system.
The prompt above was compiled from the original drawing commands and is for reference only.