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

• $x = r \cos \theta$
• $y = r \sin \theta$
• $z = z$

Given the equation: $y = x^2$

Step 1: Substitute $x$ and $y$ in terms of cylindrical coordinates.

Since $x = r \cos \theta$ and $y = r \sin \theta$, substitute these values: $r \sin \theta = (r \cos \theta)^2$

Step 2: Simplify the equation.

Expanding $(r \cos \theta)^2$: $r \sin \theta = r^2 \cos^2 \theta$

Step 3: Divide both sides by $r$ (assuming $r \neq 0$).

$\sin \theta = r \cos^2 \theta$

This is the equation in cylindrical coordinates that corresponds to the given rectangular equation $y = x^2$.

Would you like further details on this solution, or do you have any questions?

Here are some related questions you might consider:

1. How would you convert other rectangular equations like $x = y^2$ to cylindrical coordinates?
2. What happens if the equation includes $z$ in rectangular coordinates? How do we handle that in cylindrical coordinates?
3. Can you convert $x^2 + y^2 = 1$ into cylindrical coordinates?
4. How do spherical coordinates differ from cylindrical coordinates in such conversions?
5. How would you find the surface or volume bounded by a cylindrical coordinate equation?

Tip: Always double-check the transformation equations when switching between coordinate systems, as minor errors can lead to incorrect results.