We computed the volume of ball, using the area of a disk.
Then we tried to compute it using the area of a sphere.
Since we (pretended we) didn't know that area, we worked backwards, differentiating the volume formula w.r.t. radius to get the area of the sphere.
We computed the volume of a cube by slicing it into equal squares. Pretty silly.
Then we computed it again, by slicing at 45 degrees, into rectangles all of the same height. Sillier.
Then I described what the slices look like lengthwise: triangles, then hexagons, then triangles again, and left it to the masochists to confirm that the volume is still 1.
Definition: the average value of a function. Statement of the Mean Value Theorem in this context.
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