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You have found (or approximated) the volume of a rotated flag with various shapes. What if the axis of rotation (the “pole”) is not attached directly to the flag?

In the graph at right, there is a gap between the flag and the x-axis. Decide what will occur when this flag is rotated about the -axis. Draw a sketch of the result and calculate the volume of the resulting solid.

A large cylinder, with a hollow cylindrical hole in the center.

Compute two volumes and subtract: a large solid cylinder the cylindrical hole.

Both cylinders have a height of .

The large cylinder has a radius of . The small cylinder has a radius of .

First quadrant, with shaded rectangle with vertices as follows: (1, comma 1), (1, comma 2), (4, comma 2), & (4, comma 1).

Use the eTool below to visualize the flag.
Click the link at right for the full version of the eTool: Calc 2-121 HW eTool