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Calculus Course

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X-Ray OriginalRingsSlicesBoards

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Ring By Ring Timelapse 25%100%50%75%

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X-Ray

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Shape

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X-Ray ViewStep-by-step

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X-Ray Shape Step-by-step

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X-Ray StrategyVisualizationStep-by-Step PatternCalculus Lingo Ring-by-ring Slice-by-slice Board-by-board

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X-Ray StrategyVisualizationStep-by-Step LayoutCalculus Lingo Ring-by-ring Slice-by-slice Board-by-board

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X-Ray StrategyVisualizationStep-by-Step LayoutCalculus Lingo Ring-by-ring Slice-by-slice Board-by-board

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X-Ray StrategyVisualizationStep-by-Step LayoutSingle Step Zoom Ring-by-ring Slice-by-slice Board-by-board r p x dp dx dr 2*pi*r r

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X-Ray StrategyVisualizationStep-by-Step LayoutSingle Step Zoom Ring-by-ring Slice-by-slice Board-by-board r p x dp dx dr 2*pi*r r

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Shape Animation Spin

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Single Ring Multiple Rings (Area) Spin Area (Volume) Dissect Volume (Surface Area)

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Start with a ring Build out a disc Combine discs into a sphere Subtract a smaller sphere to get a “Shell”

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CircleRingsSlicesBoards SphereShellsWedgesPlates

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Ring-by-ring Timelapse Shell-by-Shell X-Ray Plate-by-plate Timelapse CircumferenceAreaVolumeSurface area ?

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Shell-by-Shell X-Ray… Take a Ring Ring-by-ring timelapse… to a circle Plate-by-plate timelapse…to a sphere Take a sphere to surface area

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= + = 2 ×

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Did the explanation work? Find a better one No Yes Aha!

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