Final answer:
In cylindrical coordinates, the equation y = x² can be expressed as r = √(x² + y²). The relationship between the rectangular and cylindrical coordinates x = r cos(θ) and y = r sin(θ) is used to derive this equation. By substituting these equations into the given equation and simplifying, we obtain the equation r = r.
Step-by-step explanation:
In cylindrical coordinates, the given equation y = x² can be expressed as r = √(x² + y²). Here, r represents the distance from the origin to the point in the xy-plane, and θ represents the angle between the positive x-axis and the line from the origin to the point.
To derive this equation, we can use the relationship between rectangular and cylindrical coordinates: x = r cos(θ) and y = r sin(θ). Substituting these equations into the given equation y = x² gives us r sin(θ) = (r cos(θ))². Simplifying leads to the equation r = √(r² cos²(θ) + r² sin²(θ)). Applying the trigonometric identity cos²(θ) + sin²(θ) = 1, we obtain r = √(r²), which simplifies to r = r.