Q: A ball of radius 17 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid.

A: 4500pi

I am mostly confused about the question...If we have a bead-like object, then how are we suppose to rotate it to make a solid? I guess I am just having a hard time visualizing this? In regards to the math part, I suppose I just need to find the volume of revolution for the cylinder and the ball separately then subtract them? Any help would be appreciated, thanks!

  • 1
    $\begingroup$ You know the volume of a sphere and a cylinder; your remaining challenge is to figure out the volume of a spherical cap; you have to subtract two volumes of those to get your answer. $\endgroup$ – J. M. is a poor mathematician Apr 20 '11 at 15:39
  • $\begingroup$ How do I do that? Subtract the volume of a similar square/2 or something? $\endgroup$ – Mr_CryptoPrime Apr 20 '11 at 15:50
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    $\begingroup$ Just to give you an idea... $\endgroup$ – J. M. is a poor mathematician Apr 20 '11 at 15:55

To set this volume up as a single integral, begin by drawing the circle of radius 17, centered at the origin in the xy-plane. Draw the line y=8. Let $R$ be the region in your circle and above the line $y=8$. Then the volume you seek is the volume of the solid of revolution created by revolving $R$ about the $x$-axis. The disk (or cross section) method works well for finding this volume. Any (modern) calculus text will cover this topic, with lots of examples.

Note you will want to find the intersection of the line $y=8$ and your circle in order to determine your limits of integration.


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