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Visual Calculus: Volumes and Areas

Visual Calculus: Volumes and Areas
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2015-06-04 00:51
FunctionCalculusIntegralGeometric ConstructionSlider Animation
This example demonstrates how to visualize volumes and areas in calculus using GeoGebra. Core math concepts include integration, summation, function graphing, and cross-sections of various geometric shapes. Construction highlights involve commands and objects like Line, Plane, Integral, TrapezoidalSum, Sequence, Segment, Sum, Polygon, Function. Suitable for high school and above students to learn and understand volume and area calculations in calculus.

GeoGebra Commands & Parameters

  1. If(showDisksHorizontal ∨ showDisksVertical ∨ showShellsHorizontal ∨ showShellsVertical, (b - a) / 50, (b - a) / n) → dx
  2. If(showInverse, Sequence[(f(i), i), i, a + dx / 2, b - dx / 2, dx], Sequence[(i, f(i)), i, a + dx / 2, b - dx / 2, dx]) → areaMRAMPointsF_{all}
  3. If(showInverse, Sequence[(f(i), i), i, a, b, dx], Sequence[(i, f(i)), i, a, b, dx]) → areaTRAPPointsF_{all}
  4. If(showInverse, Sequence[(g(i), i), i, a, b, dx], Sequence[(i, g(i)), i, a, b, dx]) → areaTRAPPointsG_{all}
  5. Line((0, 0), (1, 1)) → reflectionLine
  6. If(showInverse, Sequence[Mirror[Polygon[(i, g(i)), (i, f(i)), (i + dx, f(i)), (i + dx, g(i))], reflectionLine], i, a, b, dx], Sequence[Polygon[(i, g(i)), (i, f(i)), (i + dx, f(i)), (i + dx, g(i))], i, a, b - dx, dx]) → areaLRAM_{all}
  7. If(showInverse, Sequence[(f(i), i, 0), i, a, b, dx], Sequence[(i, f(i), 0), i, a, b, dx]) → pointF
  8. If(showInverse, Sequence[(f(i), i, abs(f(i) - g(i))), i, a, b, dx], Sequence[(i, f(i), abs(f(i) - g(i))), i, a, b, dx]) → pointF_{square}
  9. If(showInverse, Sequence[(g(i), i, abs(f(i) - g(i))), i, a, b, dx], Sequence[(i, g(i), abs(f(i) - g(i))), i, a, b, dx]) → pointG_{square}
  10. If(showInverse, Sequence[(g(i), i, 0), i, a, b, dx], Sequence[(i, g(i), 0), i, a, b, dx]) → pointG
  11. Zip(Polygon[A, B, C, D], A, pointF, B, pointG, C, pointG_{square}, D, pointF_{square}) → squareSection_{all}
  12. Plane((0, 0, 0), (1, 0, 0), (0, 0, 1)) → xOzPlane
  13. Plane((0, 0, 0), (0, 0, 1), (0, 1, 0)) → yOzPlane
  14. If(showInverse, Sequence[((g(i) - f(i)) / 2 + f(i), i, abs(f(i) - g(i)) / 2), i, a, b, dx], Sequence[(i, (g(i) - f(i)) / 2 + f(i), abs(f(i) - g(i)) / 2), i, a, b, dx]) → pointD_{top}
  15. If(showInverse, Sequence[((g(i) - f(i)) / 2 + f(i), i, ((-abs(f(i) - g(i)))) / 2), i, a, b, dx], Sequence[(i, (g(i) - f(i)) / 2 + f(i), ((-abs(f(i) - g(i)))) / 2), i, a, b, dx]) → pointD_{bottom}
  16. If(showInverse, Sequence[((g(i) - f(i)) / 2 + f(i), i, (abs(f(i) - g(i)) / 2 * sqrt(3))), i, a, b, dx], Sequence[(i, (g(i) - f(i)) / 2 + f(i), (abs(f(i) - g(i)) / 2 * sqrt(3))), i, a, b, dx]) → pointE
  17. If(showInverse, Sequence[(f(i), i, h(i)), i, a, b, dx], Sequence[(i, f(i), h(i)), i, a, b, dx]) → pointF_{rect}
  18. If(showInverse, Sequence[(g(i), i, h(i)), i, a, b, dx], Sequence[(i, g(i), h(i)), i, a, b, dx]) → pointG_{rect}
  19. Mirror(f, reflectionLine) → f_1
  20. Mirror(g, reflectionLine) → g_1
  21. Textfield(function_f) → inputFunction_f
  22. Textfield(function_g) → inputFunction_g
  23. Line((revAxisXequals, 0), (revAxisXequals, 1)) → revolutionAxis_v
  24. Line((0, revAxisYequals), (1, revAxisYequals)) → revolutionAxis_h
  25. Textfield(revAxisXequals) → inputXaxis
  26. Textfield(revAxisYequals) → inputYaxis
  27. Mirror(h, reflectionLine) → h_1
  28. Textfield(function_h) → inputFunction_h
  29. Element(squareSection_{all}, dxNum + 1) → highlightSquare
  30. Integral(f(x) - g(x), a, b) → sumIntegral
  31. TrapezoidalSum(f(x) - g(x), a, b, n) → sumTRAP
  32. Zip(Polygon[A, B, C], A, pointG, B, pointF, C, pointE) → equilateralSection_{all}
  33. If(showInverse, Sequence[Polygon[((f(i) + g(i)) / 2, i, (abs(f(i) - g(i)) / 2 * sqrt(3))), ((f(i) + g(i)) / 2 + (abs(f(i) - g(i)) * 3 / 4), i, (((-abs(f(i) - g(i)))) / 4 * sqrt(3))), ((f(i) + g(i)) / 2 - (abs(f(i) - g(i)) * 3 / 4), i, (((-abs(f(i) - g(i)))) / 4 * sqrt(3)))], i, a, b, dx], Sequence[Polygon[(i, (f(i) + g(i)) / 2, (abs(f(i) - g(i)) / 2 * sqrt(3))), (i, (f(i) + g(i)) / 2 + (abs(f(i) - g(i)) * 3 / 4), (((-abs(f(i) - g(i)))) / 4 * sqrt(3))), (i, (f(i) + g(i)) / 2 - (abs(f(i) - g(i)) * 3 / 4), (((-abs(f(i) - g(i)))) / 4 * sqrt(3)))], i, a, b, dx]) → equilateralSectionCentroid_{all}
  34. If(showInverse, Sequence[Polygon[((f(i) + g(i)) / 2, i, abs(f(i) - g(i)) / 2), ((abs(f(i) - g(i)) * 3 / 4) + (f(i) + g(i)) / 2, i, ((-abs(f(i) - g(i)))) / 4), (((-abs(f(i) - g(i))) * 3 / 4) + (f(i) + g(i)) / 2, i, ((-abs(f(i) - g(i)))) / 4)], i, a, b, dx], Sequence[Polygon[(i, (f(i) + g(i)) / 2, abs(f(i) - g(i)) / 2), (i, ((-abs(f(i) - g(i))) * 3 / 4) + (f(i) + g(i)) / 2, ((-abs(f(i) - g(i)))) / 4), (i, (abs(f(i) - g(i)) * 3 / 4) + (f(i) + g(i)) / 2, ((-abs(f(i) - g(i)))) / 4)], i, a, b, dx]) → isoscelesSectionCentroid_{all}
  35. Zip(Polygon[A, B, C, D], A, pointF, B, pointG, C, pointG_{rect}, D, pointF_{rect}) → rectangleSection_{all}
  36. If(showInverse, Sequence[Polygon[(f(i), i, ((-h(i))) / 2), (f(i), i, h(i) / 2), (g(i), i, h(i) / 2), (g(i), i, ((-h(i))) / 2)], i, a, b, dx], Sequence[Polygon[(i, f(i), ((-h(i))) / 2), (i, f(i), h(i) / 2), (i, g(i), h(i) / 2), (i, g(i), ((-h(i))) / 2)], i, a, b, dx]) → rectangleSectionCentroid_{all}
  37. If(showInverse, Sequence[Polygon[(f(i), i, ((-abs(f(i) - g(i)))) / 2), (f(i), i, abs(f(i) - g(i)) / 2), (g(i), i, abs(f(i) - g(i)) / 2), (g(i), i, ((-abs(f(i) - g(i)))) / 2)], i, a, b, dx], Sequence[Polygon[(i, f(i), ((-abs(f(i) - g(i)))) / 2), (i, f(i), abs(f(i) - g(i)) / 2), (i, g(i), abs(f(i) - g(i)) / 2), (i, g(i), ((-abs(f(i) - g(i)))) / 2)], i, a, b, dx]) → squareSectionCentroid_{all}
  38. Sequence(Element[isoscelesSectionCentroid_{all}, i], i, 1, Length[isoscelesSectionCentroid_{all}], 2) → isoscelesSectionCentroid_1
  39. Sequence(Element[isoscelesSectionCentroid_{all}, i], i, 2, Length[isoscelesSectionCentroid_{all}], 2) → isoscelesSectionCentroid_2
  40. Sequence(Element[rectangleSectionCentroid_{all}, i], i, 1, Length[rectangleSectionCentroid_{all}], 2) → rectangleSectionCentroid_1
  41. Sequence(Element[rectangleSectionCentroid_{all}, i], i, 2, Length[rectangleSectionCentroid_{all}], 2) → rectangleSectionCentroid_2
  42. Sequence(Element[squareSectionCentroid_{all}, i], i, 2, Length[squareSectionCentroid_{all}], 2) → squareSectionCentroid_2
  43. Sequence(Element[areaMRAMPointsF_{all}, i], i, 1, Length[areaMRAMPointsF_{all}], 2) → areaMRAMPointsF_1
  44. Sequence(Element[areaTRAPPointsG_{all}, i], i, 1, Length[areaTRAPPointsG_{all}], 2) → areaTRAPPointsG_1
  45. Sequence(Element[areaTRAPPointsF_{all}, i], i, 2, Length[areaTRAPPointsF_{all}], 2) → areaTRAPPointsF_2
  46. Sequence(Element[areaTRAPPointsG_{all}, i], i, 2, Length[areaTRAPPointsG_{all}], 2) → areaTRAPPointsG_2
  47. Sequence(If[(f(i) - g(i) < 0) ∧ (b > a) ∨ (f(i) - g(i) > 0) ∧ (b < a), Polygon[(i, g(i)), (i, f(i)), (i + (b - a) / 500, f(i + (b - a) / 500)), (i + (b - a) / 500, g(i + (b - a) / 500))]], i, a, b - (b - a) / 500, (b - a) / 500) → areaFTC_{negative}
  48. Sequence(If[(f(i) - g(i) > 0) ∧ (b > a) ∨ (f(i) - g(i) < 0) ∧ (b < a), Polygon[(i, g(i)), (i, f(i)), (i + (b - a) / 500, f(i + (b - a) / 500)), (i + (b - a) / 500, g(i + (b - a) / 500))]], i, a, b - (b - a) / 500, (b - a) / 500) → areaFTC_{positive}
  49. Mirror(areaFTC_{positive}, reflectionLine) → areaFTCInverse_{positive}
  50. Mirror(areaFTC_{negative}, reflectionLine) → areaFTCInverse_{negative}
  51. Sequence((i, f(i)), i, a, b, (2 * dx)) → arcLengthPoints_1
  52. Sequence((i, f(i)), i, a + dx, b, (2 * dx)) → arcLengthPoints_2
  53. Mirror(arcLengthPoints_1, reflectionLine) → arcLengthInversePoints_1
  54. Mirror(arcLengthPoints_2, reflectionLine) → arcLengthInversePoints_2
  55. Sequence(Segment[(i, f(i)), (i + dx, f(i + dx))], i, a, b - dx, (2 * dx)) → arcLengthSegments_1
  56. Sequence(Segment[(i, f(i)), (i + dx, f(i + dx))], i, a + dx, b - dx, (2 * dx)) → arcLengthSegments_2
  57. Mirror(arcLengthSegments_2, reflectionLine) → arcLengthInverseSegments_2
  58. Mirror(arcLengthSegments_1, reflectionLine) → arcLengthInverseSegments_1
  59. If(showInverse, Sequence[Circle[((f(i) + g(i)) / 2, i, 0), abs(f(i) - g(i)) / 2, yAxis], i, a, b, dx], Sequence[Circle[(i, (f(i) + g(i)) / 2, 0), abs(f(i) - g(i)) / 2, xAxis], i, a, b, dx]) → circleSection_{all}
  60. If(showInverse, (g(dxBar), dxBar), (dxBar, g(dxBar))) → dxAnchor
  61. If(showInverse, Segment[(f(dxBar), dxBar), (g(dxBar), dxBar)], Segment[(dxBar, f(dxBar)), (dxBar, g(dxBar))]) → highlightSegmentBottom
  62. Element(equilateralSectionCentroid_{all}, dxNum + 1) → highlightEquilateralCentroid
  63. Segment((dxBar, f(dxBar)), (dxBar + dx, f(dxBar + dx))) → highlightArcLengthSegment
  64. Segment((f(dxBar), dxBar), (f(dxBar + dx), dxBar + dx)) → highlightArcLengthSegmentInverse
  65. Zip(Polygon[A, B, C, D], A, pointF, B, pointD_{top}, C, pointG, D, pointD_{bottom}) → diamondSection_{all}
  66. Mirror(highlightSegmentBottom, revolutionAxis_h) → highlightSegmentBottom_h
  67. Mirror(highlightSegmentBottom, revolutionAxis_v) → highlightSegmentBottom_v
  68. Segment((dxBar, f(dxBar)), Mirror[(dxBar, f(dxBar)), revolutionAxis_h]) → highlightRevolutionX_f
  69. Segment((dxBar, g(dxBar)), Mirror[(dxBar, g(dxBar)), revolutionAxis_h]) → highlightRevolutionX_g
  70. Segment((dxBar, f(dxBar)), Mirror[(dxBar, f(dxBar)), revolutionAxis_v]) → highlightShellsX_f
  71. Segment((dxBar, g(dxBar)), Mirror[(dxBar, g(dxBar)), revolutionAxis_v]) → highlightShellsX_g
  72. Segment((f(dxBar), dxBar), Mirror[(f(dxBar), dxBar), revolutionAxis_v]) → highlightRevolutionY_f
  73. Segment((g(dxBar), dxBar), Mirror[(g(dxBar), dxBar), revolutionAxis_v]) → highlightRevolutionY_g
  74. Segment((f(dxBar), dxBar), Mirror[(f(dxBar), dxBar), revolutionAxis_h]) → highlightShellsY_f
  75. Segment((g(dxBar), dxBar), Mirror[(g(dxBar), dxBar), revolutionAxis_h]) → highlightShellsY_g
  76. Textfield(n) → inputN
  77. Sum(Sequence[(dx * (f(i) - g(i))), i, a, b - dx, dx]) → sumLRAM
  78. Sum(Sequence[(dx * (f(i) - g(i))), i, a + dx / 2, b, dx]) → sumMRAM
  79. Sum(Sequence[(dx * (f(i) - g(i))), i, a + dx, b, dx]) → sumRRAM
  80. Polygon((a, 0), (b, 0), (b, averageValue), (a, averageValue)) → averageValueRectangle, averageValueSegment_2, averageValueSegment_1, averageValueSegment_3, averageValueSegment_4
  81. Mirror(averageValueRectangle, reflectionLine) → averageValueRectangleInverse
  82. Textfield(a) → boundLeft
  83. Textfield(b) → boundRight
  84. Function(f, a, b) → f2
  85. Function(g, a, b) → g2
  86. Mirror(f2, reflectionLine) → f_{reflected}2
  87. Mirror(g2, reflectionLine) → g_{reflected}2
  88. Polygon(PointList[Union[Sequence[{i, f(i)}, i, a, b, (b - a) / 100], Sequence[{i, g(i)}, i, b, a, (a - b) / 100]]]) → functionBase
  89. Mirror(functionBase, reflectionLine) → functionBaseInverse
  90. Mirror(functionBaseInverse, revolutionAxis_h) → functionBaseInverse_h
  91. Mirror(functionBaseInverse, revolutionAxis_v) → functionBaseInverse_v
  92. Mirror(functionBase, revolutionAxis_h) → functionBase_h
  93. Mirror(functionBase, revolutionAxis_v) → functionBase_v
  94. Integral(abs(f(x) - g(x))^(2), a, b) → integralFminusGsquared
  95. Integral(abs((h(x) * (f(x) - g(x)))), a, b) → integralRectangle
  96. Integral(abs(f(x) - revAxisYequals)^(2) - abs(g(x) - revAxisYequals)^(2), a, b) → volumeDisks_{dx}
  97. Integral(abs(f(x) - revAxisXequals)^(2) - abs(g(x) - revAxisXequals)^(2), a, b) → volumeDisks_{dy}
  98. Integral((abs(x - revAxisXequals) * (f(x) - g(x))), a, b) → volumeShells_{dx}
  99. Integral((abs(x - revAxisYequals) * (f(x) - g(x))), a, b) → volumeShells_{dy}
  100. Integral((((2 * π) * (f(x) - revAxisYequals)) * sqrt(1 + Derivative[f]^(2))), a, b) → surfaceArea_x
  101. Integral((((2 * π) * (f(x) - revAxisXequals)) * sqrt(1 + Derivative[f]^(2))), a, b) → surfaceArea_y
  102. Integral(sqrt(1 + Derivative[f]^(2)), a, b) → arcLength_f
  103. SelectedIndex(exampleList) → exampleNumber
  104. Sequence(Element[squareSectionCentroid_{all}, i], i, 3, Length[squareSectionCentroid_{all}], 2) → squareSectionCentroid_1
  105. Sequence(Element[circleSection_{all}, i], i, 2, Length[circleSection_{all}], 2) → circleSection
  106. Sequence(Element[circleSection_{all}, i], i, 1, Length[circleSection_{all}], 2) → circleSection_2
  107. Sequence(Element[diamondSection_{all}, i], i, 2, Length[diamondSection_{all}], 2) → diamondSection_1
  108. Sequence(Element[diamondSection_{all}, i], i, 1, Length[diamondSection_{all}], 2) → diamondSection_2
  109. Sequence(Element[equilateralSectionCentroid_{all}, i], i, 2, Length[equilateralSectionCentroid_{all}], 2) → equilateralSectionCentroid_1
  110. Sequence(Element[equilateralSectionCentroid_{all}, i], i, 1, Length[equilateralSectionCentroid_{all}], 2) → equilateralSectionCentroid_2
  111. Sequence(Element[equilateralSection_{all}, i], i, 1, Length[equilateralSection_{all}], 2) → equilateralSection_2
  112. Sequence(Element[equilateralSection_{all}, i], i, 2, Length[equilateralSection_{all}], 2) → equilateralSection_{1}
  113. Sequence(Element[rectangleSection_{all}, i], i, 2, Length[rectangleSection_{all}], 2) → rectangleSection_1
  114. Sequence(Element[rectangleSection_{all}, i], i, 1, Length[rectangleSection_{all}], 2) → rectangleSection_2
  115. Sequence(Element[squareSection_{all}, i], i, 1, Length[squareSection_{all}], 2) → squareSection_2
  116. Sequence(Element[squareSection_{all}, i], i, 2, Length[squareSection_{all}], 2) → squareSection_1
  117. Zip(Polygon[A, B, C], A, pointG, B, pointF, C, pointF_{square}) → isoscelesSectionF_{all}
  118. Zip(Polygon[A, B, C], A, pointF, B, pointG, C, pointG_{square}) → isoscelesSectionG_{all}
  119. Sequence(Element[isoscelesSectionG_{all}, i], i, 1, Length[isoscelesSectionG_{all}], 2) → isoscelesSectionG_2
  120. Sequence(Element[isoscelesSectionG_{all}, i], i, 2, Length[isoscelesSectionG_{all}], 2) → isoscelesSectionG_1
  121. Sequence(Element[isoscelesSectionF_{all}, i], i, 2, Length[isoscelesSectionF_{all}], 2) → isoscelesSectionF_2
  122. Sequence(Element[isoscelesSectionF_{all}, i], i, 1, Length[isoscelesSectionF_{all}], 2) → isoscelesSectionF_1
  123. If(showInverse, Sequence[Segment[(g(i), i), (f(i), i)], i, a, b, dx], Sequence[Segment[(i, g(i)), (i, f(i))], i, a, b, dx]) → segmentBottom_{all}
  124. Sequence(Element[segmentBottom_{all}, i], i, 1, Length[segmentBottom_{all}], 2) → segmentBottom_2
  125. Sequence(Element[segmentBottom_{all}, i], i, 2, Length[segmentBottom_{all}], 2) → segmentBottom_1
  126. If(showInverse, Sequence[Sector[Circle[((f(i) + g(i)) / 2, i, 0), abs(f(i) - g(i)) / 2, xOzPlane], 0, 3.141592653589793], i, a, b, dx], Sequence[Sector[Circle[(i, (f(i) + g(i)) / 2, 0), abs(f(i) - g(i)) / 2, yOzPlane], 0, 3.141592653589793], i, a, b, dx]) → semiCircleSection_{all}
  127. Sequence(Element[semiCircleSection_{all}, i], i, 2, Length[semiCircleSection_{all}], 2) → semiCircleSection_1
  128. Sequence(Element[semiCircleSection_{all}, i], i, 1, Length[semiCircleSection_{all}], 2) → semiCircleSection_2
  129. If(showInverse, Sequence[Sector[Circle[((f(i) + g(i)) / 2, i, (((-abs((f(i) - g(i)) / 2))) / sqrt(1 - 16 / ((9 * π^(2)))) * 4 / 3 / π)), abs((f(i) - g(i)) / 2) / sqrt(1 - 16 / ((9 * π^(2)))), xOzPlane], 0, 3.141592653589793], i, a, b, dx], Sequence[Sector[Circle[(i, (f(i) + g(i)) / 2, (((-abs((f(i) - g(i)) / 2))) / sqrt(1 - 16 / ((9 * π^(2)))) * 4 / 3 / π)), abs((f(i) - g(i)) / 2) / sqrt(1 - 16 / ((9 * π^(2)))), yOzPlane], 0, 3.141592653589793], i, a, b, dx]) → semiCircleSectionCentroid_{all}
  130. Sequence(Element[semiCircleSectionCentroid_{all}, i], i, 2, Length[semiCircleSectionCentroid_{all}], 2) → semiCircleSectionCentroid_1
  131. Sequence(Element[semiCircleSectionCentroid_{all}, i], i, 1, Length[semiCircleSectionCentroid_{all}], 2) → semiCircleSectionCentroid_2
  132. Element(diamondSection_{all}, dxNum + 1) → highlightDiamond
  133. Element(equilateralSection_{all}, dxNum + 1) → highlightEquilateral
  134. Element(isoscelesSectionCentroid_{all}, dxNum + 1) → higlightIsoscelesCentroid
  135. Element(isoscelesSectionG_{all}, dxNum + 1) → highlightIsosceles2_G
  136. Element(isoscelesSectionF_{all}, dxNum + 1) → highlightIsosceles2_F
  137. Element(rectangleSection_{all}, dxNum + 1) → highlightRectangle
  138. Element(rectangleSectionCentroid_{all}, dxNum + 1) → highlightRectangleCentroid
  139. Element(squareSectionCentroid_{all}, dxNum + 1) → highlightSquareCentroid
  140. Element(circleSection_{all}, dxNum + 1) → highlightSectionCircle
  141. Element(semiCircleSection_{all}, dxNum + 1) → highlightSemiCircle
  142. Element(semiCircleSectionCentroid_{all}, dxNum + 1) → highlightSemiCircleCentroid
  143. Element(areaLRAM_{all}, dxNum + 1) → highlightAreaLRAM
  144. Sequence(Element[areaLRAM_{all}, i], i, 2, Length[areaLRAM_{all}], 2) → areaLRAM_2
  145. Sequence(Element[areaLRAM_{all}, i], i, 1, Length[areaLRAM_{all}], 2) → areaLRAM_1
  146. If(showInverse, Sequence[Mirror[Polygon[(i, g(i + dx / 2)), (i, f(i + dx / 2)), (i + dx, f(i + dx / 2)), (i + dx, g(i + dx / 2))], reflectionLine], i, a, b - dx, dx], Sequence[Polygon[(i, g(i + dx)), (i, f(i + dx)), (i + dx, f(i + dx)), (i + dx, g(i + dx))], i, a, b - dx, dx]) → areaRRAM_{all}
  147. Sequence(Element[areaRRAM_{all}, i], i, 1, Length[areaRRAM_{all}], 2) → areaRRAM_1
  148. Sequence(Element[areaRRAM_{all}, i], i, 2, Length[areaRRAM_{all}], 2) → areaRRAM_2
  149. Element(areaRRAM_{all}, dxNum + 1) → highlightAreaRRAM
  150. If(showInverse, Sequence[Mirror[Polygon[(i, g(i + dx / 2)), (i, f(i + dx / 2)), (i + dx, f(i + dx / 2)), (i + dx, g(i + dx / 2))], reflectionLine], i, a, b - dx, dx], Sequence[Polygon[(i, g(i + dx / 2)), (i, f(i + dx / 2)), (i + dx, f(i + dx / 2)), (i + dx, g(i + dx / 2))], i, a, b - dx, dx]) → areaMRAM_{all}
  151. Sequence(Element[areaMRAM_{all}, i], i, 1, Length[areaMRAM_{all}], 2) → areaMRAM_1
  152. Sequence(Element[areaMRAM_{all}, i], i, 2, Length[areaMRAM_{all}], 2) → areaMRAM_2
  153. Element(areaMRAM_{all}, dxNum + 1) → highlightAreaMRAM
  154. If(showInverse, Sequence[Mirror[Polygon[(i, g(i)), (i, f(i)), (i + dx, f(i + dx)), (i + dx, g(i + dx))], reflectionLine], i, a, b - dx, dx], Sequence[Polygon[(i, g(i)), (i, f(i)), (i + dx, f(i + dx)), (i + dx, g(i + dx))], i, a, b - dx, dx]) → areaTRAP_{all}
  155. Sequence(Element[areaTRAP_{all}, i], i, 1, Length[areaTRAP_{all}], 2) → areaTRAP_1
  156. Sequence(Element[areaTRAP_{all}, i], i, 2, Length[areaTRAP_{all}], 2) → areaTRAP_2
  157. Element(areaTRAP_{all}, dxNum + 1) → highlightAreaTRAP
  158. If(showInverse, Sequence[(g(i), i), i, a + dx, b, dx], Sequence[(i, g(i)), i, a + dx, b, dx]) → areaRRAMPointsG_{all}
  159. If(showInverse, Sequence[(f(i), i), i, a + dx, b, dx], Sequence[(i, f(i)), i, a + dx, b, dx]) → areaRRAMPointsF_{all}
  160. If(showInverse, Sequence[(g(i), i), i, a + dx / 2, b - dx / 2, dx], Sequence[(i, g(i)), i, a + dx / 2, b - dx / 2, dx]) → areaMRAMPointsG_{all}
  161. If(showInverse, Sequence[(f(i), i), i, a, b - dx, dx], Sequence[(i, f(i)), i, a, b - dx, dx]) → areaLRAMPointsF_{all}
  162. If(showInverse, Sequence[(g(i), i), i, a, b - dx, dx], Sequence[(i, g(i)), i, a, b - dx, dx]) → areaLRAMPointsG_{all}
  163. Sequence(Element[areaTRAPPointsF_{all}, i], i, 1, Length[areaTRAPPointsF_{all}], 2) → areaTRAPPointsF_1
  164. Sequence(Element[areaMRAMPointsF_{all}, i], i, 2, Length[areaMRAMPointsF_{all}], 2) → areaMRAMPointsF_2
  165. Sequence(Element[areaMRAMPointsG_{all}, i], i, 1, Length[areaMRAMPointsG_{all}], 2) → areaMRAMPointsG_1
  166. Sequence(Element[areaMRAMPointsG_{all}, i], i, 2, Length[areaMRAMPointsG_{all}], 2) → areaMRAMPointsG_2
  167. Sequence(Element[areaRRAMPointsF_{all}, i], i, 2, Length[areaRRAMPointsF_{all}], 2) → areaRRAMPointsF_2
  168. Sequence(Element[areaRRAMPointsF_{all}, i], i, 1, Length[areaRRAMPointsF_{all}], 2) → areaRRAMPointsF_1
  169. Sequence(Element[areaRRAMPointsG_{all}, i], i, 1, Length[areaRRAMPointsG_{all}], 2) → areaRRAMPointsG_1
  170. Sequence(Element[areaRRAMPointsG_{all}, i], i, 2, Length[areaRRAMPointsG_{all}], 2) → areaRRAMPointsG_2
  171. Sequence(Element[areaLRAMPointsG_{all}, i], i, 1, Length[areaLRAMPointsG_{all}], 2) → areaLRAMPointsG_1
  172. Sequence(Element[areaLRAMPointsG_{all}, i], i, 2, Length[areaLRAMPointsG_{all}], 2) → areaLRAMPointsG_2
  173. Sequence(Element[areaLRAMPointsF_{all}, i], i, 2, Length[areaLRAMPointsF_{all}], 2) → areaLRAMPointsF_2
  174. Sequence(Element[areaLRAMPointsF_{all}, i], i, 1, Length[areaLRAMPointsF_{all}], 2) → areaLRAMPointsF_1
  175. If(showInverse, CurveCartesian[revAxisXequals + ((f(dxBar) - revAxisXequals) * cos(v)), dxBar, (((-f(dxBar)) + revAxisXequals) * sin(v)), v, 0, rotationAngle], CurveCartesian[dxBar, revAxisYequals + ((f(dxBar) - revAxisYequals) * cos(v)), (((-f(dxBar)) + revAxisYequals) * sin(v)), v, 0, rotationAngle]) → highlightSurfaceArea
  176. Surface(revAxisXequals + ((dxBar - revAxisXequals) * cos(t)), u, ((dxBar - revAxisXequals) * sin(t)), t, 0, rotationAngle, u, g(dxBar), f(dxBar)) → highlightShellsVertical
  177. Surface(u, revAxisYequals + ((dxBar - revAxisYequals) * cos(t)), ((dxBar - revAxisYequals) * sin(t)), t, 0, rotationAngle, u, g(dxBar), f(dxBar)) → highlightShellsHorizontal
  178. Sequence(CurveCartesian[revAxisYequals + ((j - revAxisYequals) * cos(v)), dxBar, ((-((-j) + revAxisYequals)) * sin(v)), v, 0, rotationAngle], j, f(dxBar), g(dxBar), ((-(f(dxBar) - g(dxBar)))) / 25) → disksHighlight_{dy}
  179. Sequence(CurveCartesian[dxBar, revAxisYequals + ((j - revAxisYequals) * cos(v)), ((-((-j) + revAxisYequals)) * sin(v)), v, 0, rotationAngle], j, f(dxBar), g(dxBar), ((-(f(dxBar) - g(dxBar)))) / 25) → disksHighlight_{dx}
  180. Sequence(Circle[(i, revAxisYequals), abs(f(i) - revAxisYequals), xAxis], i, a, b, (b - a) / n) → disksF_{all}
  181. Sequence(Circle[(i, revAxisYequals), abs(g(i) - revAxisYequals), xAxis], i, a, b, (b - a) / n) → disksG_{all}
  182. Sequence(Element[disksF_{all}, i], i, 1, Length[disksF_{all}], 2) → disksF_1
  183. Sequence(Element[disksF_{all}, i], i, 2, Length[disksF_{all}], 2) → disksF_2
  184. Sequence(Element[disksG_{all}, i], i, 1, Length[disksG_{all}], 2) → disksG_1
  185. Sequence(Element[disksG_{all}, i], i, 2, Length[disksG_{all}], 2) → disksG_2
  186. Sequence(Circle[(revAxisXequals, i), abs(f(i) - revAxisXequals), yAxis], i, a, b, (b - a) / n) → disksInverseF_{all}
  187. Sequence(Circle[(revAxisXequals, i), abs(g(i) - revAxisXequals), yAxis], i, a, b, (b - a) / n) → disksInverseG_{all}
  188. Sequence(Element[disksInverseF_{all}, i], i, 1, Length[disksInverseF_{all}], 2) → disksInverseF_1
  189. Sequence(Element[disksInverseF_{all}, i], i, 2, Length[disksInverseF_{all}], 2) → disksInverseF_2
  190. Sequence(Element[disksInverseG_{all}, i], i, 1, Length[disksInverseG_{all}], 2) → disksInverseG_1
  191. Sequence(Element[disksInverseG_{all}, i], i, 2, Length[disksInverseG_{all}], 2) → disksInverseG_2
  192. Sequence(QuadricSide[Cylinder[(revAxisXequals, f(i)), (revAxisXequals, g(i)), i - revAxisXequals]], i, a, b, (b - a) / n) → shells_{all}
  193. Sequence(Element[shells_{all}, i], i, 1, Length[shells_{all}], 2) → shells_1
  194. Sequence(Element[shells_{all}, i], i, 2, Length[shells_{all}], 2) → shells_2
  195. Sequence(QuadricSide[Cylinder[(f(i), revAxisYequals), (g(i), revAxisYequals), i - revAxisYequals]], i, a, b, (b - a) / n) → shellsInverse_{all}
  196. Sequence(Element[shellsInverse_{all}, i], i, 1, Length[shellsInverse_{all}], 2) → shellsInverse_1
  197. Sequence(Element[shellsInverse_{all}, i], i, 2, Length[shellsInverse_{all}], 2) → shellsInverse_2
  198. If(showInverse, Sequence[Segment[(f(i), i), (g(i), i)], i, a, b, dx], Sequence[Segment[(i, f(i)), (i, g(i))], i, a, b, dx]) → segmentSection
  199. Element(segmentSection, dxNum + 1) → highlightSegmentSections2

AI Drawing Prompt Reference

Visualize calculus concepts by drawing two function graphs and the area between them. Use left and right Riemann sums, as well as trapezoidal sums, to approximate the area. Include mirrored images of the functions and geometric shapes like squares, equilateral triangles, and rectangles derived from the reflections. These shapes represent cross-sections for volume and area calculations. Highlight the integration process with these geometric figures.
The prompt above was compiled from the original drawing commands and is for reference only.