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##### Calculus Course

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**X-Ray**Original Rings Slices Boards**Ring By Ring Timelapse**25% 50% 75% 100%**X-Ray View**Step-by-step**X-Ray Shape**Step-by-step**2*pi*r**dr r r p dp dx x**2*pi*r**dr r r p dp dx x**Start with a ring**Build out a disc Combine discs into a sphere Subtract a smaller sphere to get a “Shell”**Circle**Rings Slices Boards Sphere Shells Wedges Plates**?**Circumference Area Volume Surface area Ring-by-ringTimelapse Plate-by-plateTimelapse Shell-by-ShellX-Ray**Take a Ring**Ring-by-ring timelapse… to a circle Plate-by-plate timelapse… to a sphere Shell-by-Shell X-Ray… to surface area Take a sphere**=**+ = • 2 ×**No**• Aha! Yes