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To convert the given rectangular equation \( x = y^2 \) into cylindrical coordinates, we use the standard transformations between rectangular \((x, y, z)\) and cylindrical coordinates \((r, \theta, z)\). The transformations are:

- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)

Given the equation \( x = y^2 \), we now substitute the cylindrical equivalents of \(x\) and \(y\):

1. Substituting \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\):
   \[
   r \cos(\theta) = (r \sin(\theta))^2
   \]
   
2. Expanding the right-hand side:
   \[
   r \cos(\theta) = r^2 \sin^2(\theta)
   \]

3. Simplifying the equation:
   \[
   \cos(\theta) = r \sin^2(\theta)
   \]
   This is the equation in cylindrical coordinates.

Let me know if you'd like more details on this transformation or any other clarifications!

Here are 5 related questions:
1. How do you convert other types of rectangular equations into cylindrical coordinates?
2. What are the cylindrical coordinate equivalents for higher powers of \(x\) or \(y\)?
3. How do cylindrical coordinates relate to spherical coordinates?
4. Can you find an inverse transformation from cylindrical back to rectangular coordinates?
5. How do you handle 3D surfaces when converting to cylindrical coordinates?

**Tip:** When working with cylindrical coordinates, always pay close attention to trigonometric identities, as they often simplify the final form of the equation.

Find an equation in cylindrical coordinates for the rectangular equation x = y^2.

Learn how to convert the rectangular equation x = y^2 into cylindrical coordinates using trigonometric transformations. This step-by-step guide covers key concepts in cylindrical coordinate transformations. Ideal for students in Grades 11-12.

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