A hole 6 inches long is drilled through a sphere of radius R to form a ring. (See the figure below; the ring is 6 inches high and 10 inches across, and R is 5 inches.)

Find an expression for the volume of the ring. Is there anything remarkable about this result?

Hint 1

The ring is a volume of revolution.

Hint 2

You can find the volume by the shell method or by the disk method, but the disk method is easier. Turn the figure on its side, so we are rotating about the x axis.

The ring is formed by rotating the area bounded by the semi-circle x² + y² = R², y >= 0 and the line of width 6 that intersects it, about the x axis. Let's write y_c for the y value of the circle, and y_l for the y value of the straight line. Then the volume of the ring is

Now, what is surprising about this result?

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