Final answer:
To find the volume of the solid E, we need to integrate over the given boundaries in rectangular coordinates and convert them to cylindrical coordinates. The volume of solid E is equal to the evaluated integral ∫∫ (2 - r²) rdrdθ over the region 0 ≤ r ≤ √(2) and 0 ≤ θ ≤ 2π. The answer is 2π.
Step-by-step explanation:
To find the volume of the solid E, we need to integrate over the given boundaries in rectangular coordinates. The given boundaries are x² + y² ≤ z ≤ 4 - x² - y², x ≥ 0, y ≥ 0. We can rewrite these boundaries as follows:
- x² + y² ≤ z ≤ 4 - (x² + y²)
- Simplifying further, we get 2(x² + y²) ≤ 4, which simplifies to x² + y² ≤ 2
Now, we can set up the integral for the volume as follows:
- ∫∫∫ dV = ∫∫∫ dzdydx
- Using cylindrical coordinates, we have:
- ∫∫∫ r dzdrdθ
- Since the volume of a solid E can be represented by the equation x² + y² ≤ 2, we can integrate over the region:
- 0 ≤ r ≤ √(2) and 0 ≤ θ ≤ 2π
- Thus, the volume of the solid E is given by:
- V = ∫∫∫ r dzdrdθ = ∫∫ (2 - r²) rdrdθ
Simplifying the integral and evaluating it over the given region, we can find the volume of solid E. The answer is option A. 2π.