Question

Let ℰ be the solid where x² + y² ≥ 1 and x² + y² ≤ z ≤ 4.

Set up an iterated integral to calculate.
You can easily copy and paste these questions as plain text.

Answer 1

To calculate the volume of the solid ℰ, set up the iterated integral with appropriate limits of integration for each variable. Z ranges from x² + y² to 4, while x and y range over the circle with radius 1 centered at the origin.

Step-by-step explanation:

To set up the iterated integral to calculate the volume of the solid ℰ, we need to find the limits of integration for each variable. Let's start with z. From the given conditions, we have x² + y² ≤ z ≤ 4. This means that z ranges from the lower bound of x² + y² to the upper bound of 4. For x and y, the condition x² + y² ≥ 1 can be rewritten as x² + y² = 1, which represents the circle with radius 1 centered at the origin. Therefore, x and y range over this circle.

Putting it all together, the iterated integral to calculate the volume of ℰ is:

∫14 ∫circle with radius 1 centered at the origincircle with radius 1 centered at the origin ∫x2 + y24 dz dy dx.