1. Calculus Course

2. ~

3. X-Ray OriginalRingsSlicesBoards

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

6. X-Ray

8. Shape

9. X-Ray ViewStep-by-step

10. X-Ray Shape Step-by-step

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

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

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

14. 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

15. 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

16. Shape Animation Spin

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

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

21. CircleRingsSlicesBoards SphereShellsWedgesPlates

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

23. 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

24. = + = 2 ×

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