Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ z=\sqrt{4-x^{2}-4 y^{2}} $$

Step-by-Step Solution

Answer

The region \(R\) that corresponds to the domain of the function is a disc of radius 1, centered at the origin in the xy-plane.

Step 1: Identify the Value under the Square Root

The function \(z = \sqrt{4 - x^{2} - 4y^{2}}\) attempt to find the value under the square root. It's \(4 - x^{2} - 4y^{2}\)

Step 2: Set the Value Greater Than or Equal to Zero

The value under the square root must be greater than or equal to zero. This gives \(4 - x^{2} - 4y^{2} \geq 0\). Arrange the equation to identify the region, we get \(x^{2} + y^{2} \leq 1\)

Step 3: Identify the Domain Region

The equation \(x^{2} + y^{2} \leq 1\) represents a disc of radius 1 centered at the origin in the xy-plane. This includes the points lying on the circle and the points inside the circle.