Math Solutions

(Steps Shown) Find the volume of the interior of the torus with inner radius R-r and outer radius R+r which is described

Question: Find the volume of the interior of the torus with inner radius \(R-r\) and outer radius \(R+r\) which is described by

\[\begin{aligned} x &=(R+r \cos v) \cos u \\ y &=(R+r \cos v) \sin u \\ z &=r \sin v \end{aligned}\]

where \(0 \leq u \leq 2 \pi\) and \(0 \leq v \leq 2 \pi\)

  1. Find an appropriate coordinate system (in \(3 \mathrm{D})\) to describe the interior of the torus.
  2. Find the Jacobian determinant associated with this change of coordinates, and calculate the volume.

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Deliverable: Word Document


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