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A classic exercise in third-semester calculus is to find the volume of the intersection of two cylinders. Consider two cylinders of unit radius, one cylinder having its axis along the x-axis, and the second cylinder having its axis along the y-axis. The problem is to find the volume of the intersection, that is, the volume of the region that is inside both cylinders simultaneously.
The surprise is that a square appears in the solution of this problem!
Your task is to use Maple to create graphics to help visualize the problem and then to solve the problem by showing that the volume of the intersection equals 16/3. You may want to review multiple integrals if you have not done any recently. You may use Maple to compute the integrals that arise.
Here are three sample plots that expose different aspects of the problem. Try first to reproduce these figures, and then look for your own way to illustrate the problem.
The first figure is a tube plot with the default style.
The second figure is a tube plot showing contours.
The third figure is a 3d wireframe plot.
Remark: This problem about the volume of two intersecting cylinders was solved in ancient times by the famous Archimedes of Syracuse (who lived in the third century B.C.) by a technique that anticipated the modern theory of integral calculus. The work of Archimedes containing the problem is known as The Method, but the one partial copy of The Method that survives today contains only the statement, not the solution. This copy was discovered at the beginning of the twentieth century in a palimpsest. Although the palimpsest went out of circulation for most of the century, it resurfaced at the end of the century with much fanfare and was auctioned at Christie's in 1998 for two million dollars.
Follow-up exercise
Consider three cylinders
of unit radius, one having its axis along the x-axis, one
having its axis along the y-axis, and one having its axis
along the z-axis. Use the computer to illustrate the
region of intersection of the three cylinders, and show that
the volume of the region equals 16 - 8 sqrt(2).
The Math 696 course
pages were last modified April 5, 2005. | | | Application: the birdcage |