To find an equation for the surface obtained by rotating the curve $y = \sqrt{x}$ about the $x$-axis, we use the concept of surface of revolution.

The formula for the surface area of a surface of revolution generated by rotating a curve $y = f(x)$ about the $x$-axis from $x = a$ to $x = b$ is:

$S = 2\pi \int_a^b y \, \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$

Here, we are interested in the surface, so we need to describe the surface, not the area. To do that, we'll consider a parametric representation of the surface.

Step 1: Parametrize the curve

The curve is given by $y = \sqrt{x}$. When we rotate this curve about the $x$-axis, we can describe the surface parametrically as follows:

• Let $x$ be the independent variable.
• For each $x$, the point on the surface forms a circle with radius $y = \sqrt{x}$.

Therefore, the parametric equations for the surface are:

$x = x$ $y = \sqrt{x} \cdot \cos(\theta)$ $z = \sqrt{x} \cdot \sin(\theta)$

where $\theta$ is the angular parameter that varies from $0$ to $2\pi$.

Step 2: General surface equation

To express the equation in terms of $x$, $y$, and $z$, we can eliminate $\theta$. From the parametric equations:

• $y^2 = x \cos^2(\theta)$
• $z^2 = x \sin^2(\theta)$

Adding these two equations:

$y^2 + z^2 = x (\cos^2(\theta) + \sin^2(\theta)) = x$

Thus, the equation for the surface is:

$y^2 + z^2 = x$

Final Answer:

The equation for the surface obtained by rotating the curve $y = \sqrt{x}$ about the $x$-axis is:

$y^2 + z^2 = x$

Would you like any further clarifications or details on this solution? Here are a few related questions you might find interesting:

1. How do you calculate the surface area of this surface of revolution?
2. What would the equation be if we rotated the curve around the $y$-axis instead?
3. How can you find the volume of a solid obtained by rotating a curve about the axis?
4. What is the general form of a surface of revolution equation for a curve $y = f(x)$?
5. How do you compute the arc length of the curve $y = \sqrt{x}$?

Tip: The parametric form for the surface of revolution is really helpful in understanding the 3D shape, especially when combined with the circular symmetry of rotation.