Question

Volume of a Torus } Repeat Exercise 43 for a torus formed by revolving the region bounded by the circle \(x^{2}+y^{2}=r^{2}\) about the line \(x=R,\) where \(r

Short answer

The volume of the torus is given by the formula \(V=2 \times \pi^2 \times r^2 \times (R-r)\)

Step-by-step solution

Define Variables

Define r as radius of the circle and R as the line about where the region is revolving to form the torus. Assume that r<R.

Apply Formula for Volume of Torus

Use the formula to find the volume of a torus: \(V=2 \times \pi^2 \times r^2 \times (R-r)\). This represents the volume of a donut-shaped object.

Simplify Expression

Here we do not have specific values for R and r, so the expression can't be calculated further. The final volume of the torus formed by revolving the region bounded by the circle \(x^2 + y^2 = r^2\) around the line \(x = R\) is given by \(V=2 \times \pi^2 \times r^2 \times (R-r)\)