Question

In Exercises 31-36, find an equation for the surface of revolution generated by revolving the curve in the indicated coordinate plane about the given axis. \(x y=2\) \(x y\) -plane \(x\) -axis

Short answer

The equation of the surface of revolution generated by revolving the given curve about the \(x\) -axis is \(r = \frac{2}{x}\).

Step-by-step solution

Write down the equation for the curve

The equation for the curve in the xy-plane is given as \(x y=2\). We want to revolve this curve about the x-axis.

Convert to cylindrical coordinates

In cylindrical coordinates, we have \(r=\sqrt{y^2+z^2}\), where \(r\) is the distance from the origin to the curve, and \(y\) and \(z\) are the aerial coordinates. Also, in the xy-plane, \(z = 0\). Substituting \(y\) with \(r\) into the original equation of the curve, we get \(x r=2\).

Solve for r

To have an equation for the new surface of revolution, solve above equation for \(r\). We get \(r = \frac{2}{x}\).

Get the final equation

As a result, the equation of the surface of revolution generated by revolving the curve \(x y=2\) about the \(x\) -axis is \(r = \frac{2}{x}\).