Question

Describe the set in cylindrical coordinates. x²+y²≤16,x≥y (Enter your answers as exact numbers. Use symbolic notation and fractions where needed. For θ give your answer as an interval. Use ∪ for combining intervals, and an appropriate type of paranthesis "(",")","[","]" depending on whether the interval is open or closed.)

Answer 1

The set in cylindrical coordinates is r² ≤ 16, representing a cylinder of radius 4 units along the z-axis.

Step-by-step explanation:

The given set in cylindrical coordinates is defined by the inequality x²+y²≤16 and x≥y. In cylindrical coordinates, we can represent x as r cos θ and y as r sin θ, where r is the radial coordinate and θ is the azimuthal angle. So, substituting these values, the inequality becomes:

r² cos² θ + r² sin² θ ≤ 16

Simplifying further, we get:

r² ≤ 16 / (cos² θ + sin² θ)

As cos² θ + sin² θ = 1, the inequality simplifies to:

r² ≤ 16

So, the set in cylindrical coordinates is r² ≤ 16, which represents a cylinder of radius 4 units with the axis along the z-axis.