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. $$ z^{2}=4 y \quad y z \text { -plane } \quad y \text { -axis } $$

Short answer

The equation for the surface of revolution is \(y = \frac{x^{2}+z^{2}}{4}\).

Step-by-step solution

Convert to cylindrical coordinates

We first convert the given equation from Cartesian to cylindrical coordinates. In this case, as it's a revolution around the y-axis, the radial distance \(r\) equals to the \(x = \sqrt{x^{2}+z^{2}}\) but as we're in the yz-plane, \(x=0\). So the new coordinates are \(z\) and \(y\). The curve equation in cylindrical coordinates becomes \(z^{2}=4y\).

Parametrize the surface

Next, we parametrize the surface of revolution. As this problem revolves around the y-axis, we can parametrize by letting \(r = z\) and \(y = y\). This gives \(y = y\), \(z = r\), and \(x = r cos \theta\), with \(\theta\) running from \(0\) to \(2\pi\) to generate a full revolution.

Find the equation of the surface

By replacing \(r\) with \(z\) in the original equation, we obtain the equation for the surface. Substituting \(z = r\) into the original equation, \(z^{2}=4y\), we get \(r^{2}=4y\) or \(y = \frac{r^{2}}{4}\). Therefore, the equation of the surface of revolution is \(y = \frac{x^{2}+z^{2}}{4}\).