When I first started learning, I was somewhat confused by problems like "rotate a certain curve around the $x$ axis." This article will provide a detailed explanation with illustrations to clarify this issue for readers.

Such problems may require finding the surface equations formed by rotating parabolas or hyperbolas around a certain axis.

When a curve rotates around the x-axis, the $y$ and $z$ coordinates of each point will form the equation of a circle, with a radius equal to the absolute value of the original $y$ coordinate. In other words:

$y^2 + z^2 = r^2$

How do we determine $r$? Since the curve lies in the $xOy$ plane, the $y$ value of any point on the curve at the start of the rotation is the radius. Thus, we have:

$y\_1^2 + z\_1^2 = y^2$

Therefore, we only need to replace $y$ in the curve with $y^2 + z^2$ to derive the new surface equation.

Similarly, when rotating around the y-axis, the value of $x$ is the radius, so we have:

$y\_1^2 + z\_1^2 = x^2$

Note that it is important to identify which plane the curve is located in before determining the radius. For curves in the $xOz$ plane, the replacement equation is slightly different. This will not be demonstrated here:

• Rotating around the $x$ axis: $z^2 = z^2 + y^2$
• Rotating around the $y$ axis: $x^2 = x^2 + z^2$
• Rotating around the $z$ axis: $x^2 = x^2 + y^2$

This image demonstrates the rotation of a parabola in the $xOy$ plane around the x-axis by 180 degrees. The right side shows the front view, and it is easy to see:

$r^2 = y^2 = 4$

To make it easier to remember the value of $r$, you can understand it this way: To choose one of the three values $x, y, z$ as $r$, simply remove the coordinate axis around which you are rotating, and then select the intersection of the remaining two values with the two values of the plane.

For example, if a curve in the $xOy$ plane rotates around the $x$ axis, you remove the x-axis, leaving y and z. Then, find the intersection of y and z with x and y to determine that $r$ is $y$.