Problem 56 - Calculus of a Single Variable

Question

A torus is formed by revolving the graph of \((x-1)^{2}+y^{2}=1\) about the \(y\) -axis. Find the surface area of the torus.

Step-by-Step Solution

Answer

The surface area of the torus is \(2\pi^2\) square units.

Step 1: Identify the values of R and r

For the given equation, the circle that generates the torus has center at (1,0) and radius 1. Thus the larger radius \(R\) is the x-coordinate of the circle center, which is one and the smaller radius \(r\) is the radius of the circle, which is also one.

Step 2: Substitute the values into the formula

Now substitute the identified values \(R=1\) and \(r=1\) into the formula for the surface area of a torus \(A=2\pi^2 Rr\). Giving, \(A=2\pi^2 (1)(1)\).

Step 3: Simplify the expression

Simplifying the expression gives \(A=2\pi^2\).