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Use a triple integral in spherical coordinates to find the volume, $V$, of a cored apple, which consists of a sphere of radius $2$, $x^2 + y^2 + z^2 = 4$, and a cylindrical hole of radius one, $x^2 + y^2 = 1$. In other words, find the volume of the sphere with the cylinder removed.

I know that $\theta$ goes from $0$ to $2\pi$ since the sphere is complete. What I don't understand is how to:

1) Convert rectangular coordinates to spherical coordinates. My textbook gives a terrible explanation.

2) Find the bounds of the triple integral. As I said, $\theta$ goes from $0$ to $2\pi$. I have no clue as to how to find $p$ or $\varphi$. The center being removed is also throwing me.

Could some explain as to how to go about this? I'm not necessarily looking for answer but a means to solve this problem. We have an exam next week and I would really love to be able to understand it.

  • 0
    Think about my approach below.2012-10-31
  • 0
    does the volume of this shape not need to be multiplied by 8 because there are 8 octants and the graph above in only in the first octant?2013-11-14

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